Computers make

very fast,

very accurate

mistakes.

very fast,

very accurate

mistakes.

Vibrations of a beam with the variable cross-section

Free vibrations of a free supported beam with the variable cross-section or step section are analyzed. The fundamental frequencies of the beam with the variable cross-section obtained by means of the asymptotic and numerical methods are compared. For a beam with the step section the frequency equation is obtained in the explicit form. The main purpose of the study is to find the form of the beam cross-section, for which
the fundamental frequency attains its maximum, if the mass of the beam is conserved.

Vibrations of a shell with the rectangular cross-section

Vibrations of a cylindrical shell with the rectangular cross-section, which has one edge clamped and other one joined with the plate, are studied in the work. Accuracy of the approximate solutions obtained by means of Rayleigh method is discussed.
The relationship between frequencies of the shell with and without the plate is found by means of the finite element analysis. The effect of the plate thickness on shell vibration frequencies is examined.

Free Vibration Frequencies of an Inhomogeneous Circular Thin Plate

Transverse vibrations of an inhomogeneous circular thin plate are studied in the report. Using the perturbation method, asymptotic formulas are obtained for the free vibration frequencies of a plate, whose thickness and Young's modulus depend nonlinearly on the radial coordinate. The behavior of frequencies for a plate with the fixed mass or fixed average stiffness is analyzed. For the lower free vibration frequencies the results of asymptotic and finite elements analyses are compared with the results obtained by the other authors by means of different methods. For small values of the perturbation parameter the numerical and asymptotic results are in good agreement.

Kapitza Pendulum

The motion of a rigid rod with the free upper and the clamped lower ends under vertical
vibration of the support is analyzed. The analysis of the problem yields ordinary differential
equation of the second order, which is solved numerically and by means of
the asymptotic expansion method. The influence of the initial angular velocity ω_{z} on attraction of the pendulum to the vertical equilibrium
position is studied.

Experimental investigation of aging of polymer and composite materials.

The purpose of this work is to investigation the influence of aging on the mechanical and fatigue properties of polymer and composite materials.

The work presents the results of experimental studies of deformation and climatic aging of polyurethane in experiments on fatigue, creep, as well as experiments on long-term natural aging of six brands of impact-resistant polystyrene in experiments on stretching and creep.

To describe the experimental creep curves of polyurethane and polystyrene samples after aging and without aging, a modified version of the Maxwell equation is used, written in the effective time scale.

The work presents the results of experimental studies of deformation and climatic aging of polyurethane in experiments on fatigue, creep, as well as experiments on long-term natural aging of six brands of impact-resistant polystyrene in experiments on stretching and creep.

To describe the experimental creep curves of polyurethane and polystyrene samples after aging and without aging, a modified version of the Maxwell equation is used, written in the effective time scale.

Investigation of the interaction of cold atoms in a magneto-optical trap with radiation from a femtosecond laser

The aim of this work is to study the principles of laser cooling of atoms, the design and operation of a magneto-optical trap (MOT), as well as an experimental study of the interaction of cold atoms in a MOT with a laser and a theoretical interpretation of the results.

The relevance of the work lies in the fact that the laser cooling technique allows reaching temperatures of a cloud of atoms, unattainable by other known methods. In addition to the interest in basic research on the phenomenon of cold and ultracold atoms, it is important to note practical application: since the frequencies corresponding to the intervals of the energy structure of a cold atom are determined with very high accuracy, they can be used as a standard in precision physics and metrology.

The relevance of the work lies in the fact that the laser cooling technique allows reaching temperatures of a cloud of atoms, unattainable by other known methods. In addition to the interest in basic research on the phenomenon of cold and ultracold atoms, it is important to note practical application: since the frequencies corresponding to the intervals of the energy structure of a cold atom are determined with very high accuracy, they can be used as a standard in precision physics and metrology.

Modelling of functional and mechanical behavior of porous
shape memory alloy based on the approximation of its structure by beam constructions.

In this work, a model of the functional and mechanical properties of porous shape memory alloys (SMA) is presented. The model is based on the representation of the
microstructure of the samples as beam structures. The analysis of microphotographs of porous TiNi specimens obtained by the SHS and SLM makes it possible to select sets of beams of various configurations as mathematical objects of modelling. The calculations are carried out using methods of the strength of materials and a microstructural model that allows one to describe various properties of SMA. A technique for determining the parameters characterizing the pore structure has been developed. Formulas for the calculation of the porous SMA structural elements strain are obtained.The calculation algorithm is compiled in frames of a computer program, the strain of samples is simulated, and comparison with experimental data are performed. The obtained results of modelling of the behavior of porous SMA under compression in different phase states and during the temperature changes under loading show good agreement with the available experimental data.

The motion of an Earth satellite after fixing the values of its acceleration in apogee.

When a satellite moves along an orbit its acceleration changes. Further motion in the case of constant acceleration after the moment of finding the satellite in apogee is being considered. This requirement is equivalent to second-order nonlinear non-holonomic constraint, which can be considered as a motion program, imposed to the satellite motion.

Two possible solutions of the control problem are considered based on two theories of motion of non-holonomic systems with high order constraints, developed by S.A. Zegzhda and M.P. Yushkov. According to the first theory a consistent system of differential equations with respect to unknown generalized coordinates and Lagrange multipliers is constructed. The second theory is based on applying extended Gauss principle.

Both theories used to find motion paths of Earth satellite of the system “Cosmos”, which is located on almost circular orbit, and Earth satellites of systems “Molniya” and “Tundra”, which are located on highly elliptical orbits, after fixing the values of their accelerations in apogees.

Two possible solutions of the control problem are considered based on two theories of motion of non-holonomic systems with high order constraints, developed by S.A. Zegzhda and M.P. Yushkov. According to the first theory a consistent system of differential equations with respect to unknown generalized coordinates and Lagrange multipliers is constructed. The second theory is based on applying extended Gauss principle.

Both theories used to find motion paths of Earth satellite of the system “Cosmos”, which is located on almost circular orbit, and Earth satellites of systems “Molniya” and “Tundra”, which are located on highly elliptical orbits, after fixing the values of their accelerations in apogees.

Optimization of chain line

The main geometric and mechanical properties of one of the most well-known lines in the mechanics – the chain line, are discussed in the report. This line is realized in everyday life and technology at the level of millennia in the form of threads, ropes, ropes, ropes, etc. The appearance of new materials has significantly increased the strength of these flexible power cells, so that they are increasingly used in a wide variety of fields of technology, technology and construction. The question about the best use of long flexible power elements in practical designs naturally arises. These issues are the subject of this work, which sets and solves a number of problems on the best suspension of a heavy flexible thread (imitating a power line cable). Much attention is paid to the construction and selection of the quality criterion of such suspension. The obtained results may be of practical interest for the developers and builders of power lines, as well as for students of technical universities.

Dynamic fracture of discrete systems

The paper is concerned with solving the problem of oscillations of linear series-connected oscillators under impulse loading, studying the effect of chain failure, and also comparing with a continuum solution.
The continuum model does not take into account material's discrete features which are very important in the study of dynamic fracture. This is the main problem that is solved in the paper.
The solution to this problem has become the idea of considering a simple discrete model in the form of series-connected linear oscillators.
To get a clear view of the behavior of a discrete model, firstly the problem of a single oscillator’s oscillation was solved, by the example of which, obvious differences with the continual model in behavior before fracture have already been noticed. Then the answer to the problem of oscillation of an arbitrary number of oscillators was found.
Finding a solution for a discrete model is much more difficult than for a continual model. But consideration of a discrete model, as in my paper, is preferable when studying the behavior of a material before failure, since it gives a more accurate solution.
The fracture of a chain's finite number of nodes was studied in the paper, previously only problems with an infinite number of nodes were considered. The behavior under impulse loading with zero initial conditions was studied, which was also not previously studied for the finite chain of oscillators.

A new triangle in the problems of classical mechanics and biodynamics

The problem of finding the most miscellaneous triangle, that is the most asymmetric triangle, is considered. The issues of the formation of various quality criteria related to maximizing the differences of the angles or sides of a triangle and characterizing the degree of its asymmetry are discussed. A detailed analysis of both additive and multiplicative criteria is carried out, during which their advantages and disadvantages are revealed. It’s shown that the quality criterion based on maximizing the product of triangle sides differences is the most adequate, and a specific configuration of the triangle can be obtained as a result of its use. In addition, practical applications of the most asymmetric triangle are discussed, related both to the problem of the artificial Earth satellite passive stabilization in circular orbit in a Newtonian force field and to the biodynamics of a person’s hand. The obtained results allow us to conclude that it is advisable to use a similar multiplicative criterion in other optimization problems in mathematics and mechanics.

The Kardar—Parisi—Zhang model under the Kazantzev-Kraichnan ensemble
with quenched noise.

Ultraviolet divergences are analyzed for the growth
model under random motion of the fluid with
quenched noise. The random growth is described by the Kardar—Parisi—Zhang model.
The velocity field is modelled with the Kazantzev-Kraichnan ensemble.

Mathematical analysis of saccadic eye motion

The thesis is devoted to the study and modeling of eye movements. The first chapter is devoted to an explanation of the sensory conflict causes during movement on the dynamic bench "Khilov's swing" and analysis of the oculomotor response in this case. An integral part of this response is saccadic eye movement.

In the second chapter of the work, experimental forms of saccades are analyzed using experimental data and an algorithm for approximating saccades is constructed. Three new parameters are proposed for describing saccades.

The third chapter examines the movement of the eyeball as a solid, with a moment applied to it from the oculomotor muscles. Two kinds of models based on the performance problem are constructed:

1) a pair of muscles described by the Feldman model acts on the eyeball;

2) a force exerts on the eyeball from the side of each of the muscles of the pair. These forces are limited.

These models make it possible to obtain experimentally observed forms of saccades.

In the second chapter of the work, experimental forms of saccades are analyzed using experimental data and an algorithm for approximating saccades is constructed. Three new parameters are proposed for describing saccades.

The third chapter examines the movement of the eyeball as a solid, with a moment applied to it from the oculomotor muscles. Two kinds of models based on the performance problem are constructed:

1) a pair of muscles described by the Feldman model acts on the eyeball;

2) a force exerts on the eyeball from the side of each of the muscles of the pair. These forces are limited.

These models make it possible to obtain experimentally observed forms of saccades.

Motion equations of a system of solids in redundant coordinates

The report presents one of the possible conclusions of
the differential motion equations of a system of solids in redundant
coordinates. The position of a rigid body is determined by the
"vector coordinates": the radius vector of the mass center and
the three unit vectors of the associated coordinate system,
the axes of which are directed along the main central inertia axes
of the body. The system of motion equations for a solids chain is
constructed. Equations of motion in redundant coordinates (a special
form of differential equations) are compiled to describe the motion of a
loaded Stewart platform. "Parasitic" oscillations arising in the equilibrium
position of the platform are considered in the report.

Optimal positioning of circular parts in stamping process

...

The Basin of Attraction in the Generalized Kapitsa Problem

The basin of attraction of a stable vertical position of a rod in the
Kapitsa problem and its generalizations are considered. A long enough flexible rod with a free upper end and a
clumped lower end is shown to lose the vertical position under its own weight. The conditions at which
harmonically vertical vibrations favor the vertical position stability of a rod have recently been
obtained. The basin of attraction of a vertical position under vibrations is discussed in the case of its
instability in lack of vibrations. Firstly, the basin of attraction is found in the context of a classic
Kapitsa problem. A rigid rod with an elastically secured lower end is then studied to simulate the
problem of flexible rod. The asymptotic method of two-scale expansions is also used. It has been
established that the transition into a vertical position depends on the initial phase of perturbation. The basin
of attraction is found to consist of two parts. In one of them, the transition into a vertical position
remains indifferent to the initial phase, whereas in another one, some domains exhibit a dependence
on the initial phase.

Application of poroelastic models for analysis
of compression experiment on myocardium element

The possibility of using a poroelastic
model to describe the myocardial element compression experiment is studied.
A linear one-dimensional model of poroelastic medium is considered,
which allows to describe hysteresis loops arising during cyclic application
of loading and unloading (preconditioning). We made a numerical calculation
in the CAE system Mathcad, where the cases of steady-state oscillations, as
well as the case of non-stationary load were considered. Due to the use of
the poroelastic model it was possible to
describe the effect of the sample height on the type of hysteresis.