Man is a slow, sloppy, and brilliant thinker;

computers are fast, accurate, and stupid.

The resonant peculiarities of the orbital motion of charged spacecraft in geophysical fields

The resonant peculiarities of the orbital motion of charged spacecraft in geophysical fields are discussed in the report. The spacecraft motion in an orbit close to circular equatorial under the Lorentz force effect is studied. The “tilted dipole” is considered as a mathematical model of the Earth magnetic field. The non-linear non-autonomous differential equation system of the spacecraft center of mass motion in the Cartesian and the spherical coordinate systems is derived. Linearized equation system of motion is investigated based on introduction of the small parameter and the method of averaging. As a result, the resonant ratios of the angular velocity of the orbital motion and the angular velocity of the Earth’s rotation are found. The existence of the special orbit for which the perturbing effect of the Lorentz force is resonant is revealed. As a result of the numerical experiment, sharp increase in the amplitude of the value of radius-vector of the spacecraft center of mass oscillations is established. Also, the increase of the special orbit inclination over time is revealed, due to which it is ceased to be equatorial. As a results, the resonant peculiarities of the orbital motion of charged spacecraft in geophysical fields are found.

The resonant peculiarities of the orbital motion of charged spacecraft in geophysical fields

The resonant peculiarities of the orbital motion of charged spacecraft in geophysical fields are discussed in the report. The spacecraft motion in an orbit close to circular equatorial under the Lorentz force effect is studied. The “tilted dipole” is considered as a mathematical model of the Earth magnetic field. The non-linear non-autonomous differential equation system of the spacecraft center of mass motion in the Cartesian and the spherical coordinate systems is derived. Linearized equation system of motion is investigated based on introduction of the small parameter and the method of averaging. As a result, the resonant ratios of the angular velocity of the orbital motion and the angular velocity of the Earth’s rotation are found. The existence of the special orbit for which the perturbing effect of the Lorentz force is resonant is revealed. As a result of the numerical experiment, sharp increase in the amplitude of the value of radius-vector of the spacecraft center of mass oscillations is established. Also, the increase of the special orbit inclination over time is revealed, due to which it is ceased to be equatorial. As a results, the resonant peculiarities of the orbital motion of charged spacecraft in geophysical fields are found.

Skateboard dynamics and control of its overclocking motions

The report discusses the dynamics and motion control of one of the most popular systems with nonholonomic constraints at present – a skateboard. A brief historical overview of the emergence and development of skateboarding is given. A calculation scheme of a skateboard is given and a mathematical model of its controlled movement is constructed, and for the sake of completeness, the motion equations of the system are derived using both the Appel equations and the Ferrers equations. Two options for the formation of controls that arise as a result of the coordinated choice of the control functions of the problem and lead to the overclocking of the skateboard are considered. A detailed analysis of the skateboard dynamics in the specified overclocking modes of its movement is carried out both in the absence and in the presence of viscous resistance from the environment. As a result, approximate solutions with retention of the necessary correction terms are found. On the basis of these solutions, a comparison of the efficiency indicators of increase in speed in the motion modes corresponding to the accepted options for the formation of control actions is made, and conclusions about the expediency of their use in practice are drawn.

SIMULATION OF MECHANISMS OF THE CABINET OF PRACTICAL MECHANICS OF
SPbSU

The Museum of the History of Physics and Mathematics of St. Petersburg
State University has a very general (about 100 items) collection of
models and mechanisms of the 19th century. These models, in their
time, were used both in research and in teaching students. Some of the
mechanisms were purchased from foreign workshops (the workshop of F.
Voigt in Berlin, the Paris and Geneva workshops) and the workshops of
St. Petersburg. Some mechanisms were created in the master's room at
St. Petersburg University by the mechanic V. Franzen. The report tells
about the work on modeling (animation) of these mechanisms. The most
suitable part of the collections for computer modeling turned out to
be rectilinear-guiding mechanisms (Peaucellier-Lipkin, Watt, Chebyshev,
Darboux and other mechanisms). When modeling, computer systems are
used: for two-dimensional modeling Maple and Mathematica, for
three-dimensional modeling - Blender. 3D model of the Roberts
rectilinear guiding mechanism, a mechanism that reproduces the rolling
of an ellipse on an ellipse, a transmission mechanism with a
hyperboloid, and others are demonstrated.

Mathematical model of non-ideal resonator of gyroscope type CVG. Development and Analysis.

The paper is devoted to development of mathematical model of non-ideal resonator of Coriolis vibratory gyroscope to determine the effect of imperfections on the output parameter - the splitting of eigenfrequency of the resonator oscillation. The resonator is a thin-walled hemispherical shell in which a standing elastic wave is excited at a frequency equal in value to the second elliptical eigenfrequency. As part of the work, we built a mathematical model of the ideal hemispherical shell and considered ways to account for imperfections in creating a model of a non-ideal resonator. The verification of the mathematical models using numerical experiment has been carried out.

Also, with the use of methods of the theory of design of experiments and methods of global sensitivity analysis, the degree of influence of changes in the input parameters of the model (imperfection parameters) on frequency splitting has been analyzed. Certain attention is paid to the evaluation of the accuracy of approximation methods used for formulating the response surface on the basis of virtual experiment data.

Also, with the use of methods of the theory of design of experiments and methods of global sensitivity analysis, the degree of influence of changes in the input parameters of the model (imperfection parameters) on frequency splitting has been analyzed. Certain attention is paid to the evaluation of the accuracy of approximation methods used for formulating the response surface on the basis of virtual experiment data.

Nonlinear dynamics of N/MEMS under laser-induced opto-thermal excitations

The work is devoted to the study of the nonlinear dynamics of parametrically excited bending oscillations of a microbeam pinched at both ends - the basic sensitive element of a promising class of microsensors of various physical quantities - under laser thermooptical action in the form of periodically generated pulses acting on a certain part of the surface of the beam element. An analytical solution of the heat conduction problem is found for a steady-state harmonic temperature distribution in the resonator volume. The static and dynamic components of force factors of temperature nature - temperature axial force and bending moment - are determined. Using the Galerkin method, the discretization of nonlinear coupled partial differential equations describing the longitudinal-flexural oscillations of the resonator is performed. Using the asymptotic method of many scales, an approximate analytical solution is obtained for the problem of nonlinear system dynamics under the conditions of the main parametric resonance. The fundamental technical feasibility of laser generation of parametric oscillations of high-Q micromechanical resonators without the implementation of scenarios for the loss of elastic stability of the sensitive element or its unacceptable heating is shown.

In addition, the results of a study of the dynamics of parametrically excited bending vibrations of two weakly coupled beam microcavities under laser thermo-optical excitation are reported. The zones of possible buildup of parametric oscillations and the amplitudes of steady-state modes for both resonators are analytically obtained. It is shown that a small difference in the mass-inertia characteristics of the resonators leads to a significant change in the amplitudes of the steady-state modes for each resonator, which can be used to detect the mass of a particle deposited on one of the sensitive elements.

In addition, the results of a study of the dynamics of parametrically excited bending vibrations of two weakly coupled beam microcavities under laser thermo-optical excitation are reported. The zones of possible buildup of parametric oscillations and the amplitudes of steady-state modes for both resonators are analytically obtained. It is shown that a small difference in the mass-inertia characteristics of the resonators leads to a significant change in the amplitudes of the steady-state modes for each resonator, which can be used to detect the mass of a particle deposited on one of the sensitive elements.

Numerical modeling of hydrodynamic accidents: erosion of dams and flooding of territories

A mathematical and numerical model of the joint dynamics of shallow waters and traction sediments is constructed, which takes into account the nonlinear dynamics of the fluid and bottom deformations. Shallow water dynamics is described by the Saint-Venant equations, taking into account the spatially inhomogeneous distribution of the terrain. The transport of entrained sediments is described by the Exner equation, generalized to the case of a spatially inhomogeneous distribution of the parameters of the underlying surface. The numerical model includes a digital terrain model (DTM) and a numerical method for integrating a system of equations describing the joint dynamics of shallow water and sediment. For the numerical integration of the Saint-Venant and Exner equations, a stable and well-tested CSPH-TVD method of the second order of accuracy is used, the parallel CUDA algorithm of which is implemented as the EcoGIS-Simulation software package for high-performance computing on supercomputers with graphics coprocessors (GPUs). Hydrodynamic modeling of the processes of erosion of the enclosing dam of a real hydrotechnical facility and flooding of adjacent territories was carried out. The parameters of the opening of the enclosing dam and flood zones, formed as a result of the development of a hydrotechnical accident at the tailing dump, were determined. Based on the obtained results, it was concluded that the method proposed in the work for numerical modeling of the joint dynamics of shallow water and traction sediments can be more versatile and efficient (has significantly better accuracy and performance) compared to existing methods for calculating the parameters of the opening and flood zones, especially for hydrodynamic currents with complex geometry on an inhomogeneous bottom topography.

Optimization issues in ballistics problems
(based on the master's thesis by D. A. Penchikov
and with the participation of B. A. Smolnikov)

А series of ballistic trajectory optimization problems is considered in the report. A presentation of the classical optimization problem of hitting a given point with a minimum initial energy is given. A two-factor criterion is constructed that combines in its structure the energy and time characteristics of the flight and allows to find the best compromise between them. An optimization criterion related to the accuracy of hitting the target is considered. The problems of flight to a maximum range with a given initial speed and of flight with a minimum initial speed to a given range are discussed, taking into account the viscous resistance from the environment. For the problem of the optimal throwing of a point in a medium with a quadratic resistance according to the criterion of the maximum flight range, the known approximate solution is compared with the results of a numerical analysis based on exact formulas. In addition, the problem of maximum range flight from a cycloidal springboard is considered, which is important for ski jumping. The obtained results are presented in a visual graphical form, and they are interesting not only from a theoretical point of view, but may also be useful in solving various applied problems.