### October 25, 2022

A.V. Lukin and L.V. Shtukin (Peter the Great St.Petersburg Polytechnic University)

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.

**
Alexei V. Lukin ** –
Associate Professor, Ph.D. High School of Mechanics and Control Processes, Physics and Mechanics Institute, Peter the Great St. Petersburg Polytechnic University. Research interests: computational mechanics, waves in continuous media, nonlinear theory of elasticity, theory of nonlinear oscillations, gyroscopy and navigation, nonlinear dynamics and control N/MEMS.

**
Lev V. Shtukin ** –
Associate Professor, Ph.D. High School of Mechanics and Control Processes, Physics and Mechanics Institute, Peter the Great St. Petersburg Polytechnic University. Research interests: computational mechanics, waves in continuous media, nonlinear theory of elasticity, theory of nonlinear oscillations, gyroscopy and navigation, nonlinear dynamics and control N/MEMS.

### October 18, 2022

### ONLINE

Khrapov S. S. (Volgograd State University, VolSU)

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.

**
Khrapov Sergey S. ** –
Associate Professor, Department of Information Systems and Computer Modeling, Volgograd State University, Ph.D.
Research interests: mathematical modeling of hydrodynamic flows, numerical methods for integrating the Saint-Venant and Exner equations, self-consistent dynamics of surface water and sediment, calculation of flood zones in case of accidents at hydraulic facilities, breakthrough and erosion of dams/dams, parallel computing on computers with graphics processors (GPUs), parallel CUDA algorithms.

### September 6, 2022

Smirnov A. S. (Peter the Great SPbPU, IPME RAS)

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.

### April 5, 2022

Zhao Shixiang (SPbSU)

Modeling of the effects of non-monotonic behavior of the yielding diagram under dynamic loading

An incremental modification of the relaxation model of plastic deformation proposed earlier has been developed in order to explain and predict the temporal effects of instability of plastic deformation diagrams. The analysis of possible scenarios in the proposed incremental model is carried out. It is shown that the incremental version of the relaxation model makes it possible to simulate the yielding diagram for long periods after the reach of the yielding point and to more fully represent the corresponding temporal effects, such as the appearance (or disappearance) of the yield drop phenomenon and the subsequent non-monotonic behaviour, including the oscillation of yielding curves. The calculations of the incremental approach are compared with the initial version of the relaxation model approach and the widespread Johnson-Cook model on the example of experimentally obtained stress-strain diagrams for different types of metals. The results confirm the descriptive and predictive capability of the incremental approach. The most important feature of the developed approach is that a limited set of parameters used in modeling the yielding curve does not depend on the history, in particular the deformation rate, and is associated only with the features of the development of the defective structure of the material at the micro- and meso-levels. Using this small set of parameters of the structural-temporal approach and the relaxation model of plasticity, it is possible to obtain various types of deformation curves for a single material in a wide range of strain rates.

**
Zhao Shixiang ** – PhD student of the Department of Theory of Elasticity of the Faculty of Mathematics and Mechanics of St. Petersburg State University. Area of scientific interests - strain rate effect, dynamic plasticity, finite element method. Scientific Advisor - Prof. Y.V. Petrov.

### March 29, 2022

N.P. Dorofeev, D.N. Ivanov, N.V. Naumova (SPbSU)

DEFORMATION OF A CIRCULAR THREE-LAYER PLATE WITH A SURFACE CHARGE

Currently, the change in the characteristics of the Solar sail is being actively investigated. The sail is usually presented as a thin film stretched over the frame. During the flight, this film undergoes deformation under the influence of a number of factors, which affects the efficiency and trajectory.

In the report, a model of the Solar sail is proposed in the form of a circular plate with a radius of 50 to 600 meters, consisting of three layers (aluminum, mylar, aluminum). A 2 ?m thick Mylar film is enclosed between two thinner aluminum films, 0.2 ?m thick.
As a result of numerical calculations in the FEM package ANSYS, the values of the maximum deflections of the plate depending on its radius were obtained. In the future, it is planned to obtain an analytical solution of the problem and to compare analytical and numerical results. The proposed model can help in predicting the motion of real Solar sails and controlling their orientation in space.

**
Dorofeev Nikita Pavlovich ** – 4th year student of the Department of Theoretical and Applied Mechanics of the Faculty of Mathematics and Mechanics of St. Petersburg State University. Area of scientific interests - deformation of multilayer plates. Scientific Advisor - Assoc. Prof. N.V. Naumova.

**
Naumova Natalia Vladimirovna ** –
Associate Professor of the Department of Theoretical and Applied Mechanics of St. Petersburg State University. Research interests: mechanics of solids, vibrations and buckling of shells, asymptotic methods for solving systems of differential equations.

**
Ivanov Denis Nikolaevich ** –
the leading engineer of the Department of Educational Process Support at St. Petersburg State University. Research interests: mechanics and mathematical modeling.

### March 22, 2022

E.A. Ivanov, G.V. Pavilainen (SPbSU)

PLASTIC ANISOTROPY IN BENDING BEAMS AND PLATES BEHIND THE ELASTIC LIMIT

The possibility of numerical and analytical solution of the problems of bending of horizontal and vertical beams, as well as circular plates made of modern structural materials with the effect of plastic anisotropy (SD effect) is analyzed. Beams and plates are subject to hydrostatic pressure and concentrated forces.

The SD effect on the symmetry breaking in the development of plasticity in bending beyond the elastic limit is shown, and the curvature of the neutral axis of the beam and the neutral surface of the plate is estimated. The comparison of analytical and numerical solutions obtained with the help of FEM in ANSYS and COMSOL packages is carried out.

**
Ivanov Evgeniy Alexandrovich ** – 2nd year Master's student of the Department of Theoretical Mechanics of the Faculty of Mathematics and Mechanics of St. Petersburg State University. Area of ??expertise: studies of stress-strain state of beams and plates under bending beyond the limits of elasticity.

**
Pavilainen Galina Voldemarovna ** –
Associate Professor of the Department of Theoretical and Applied Mechanics, St. Petersburg State University. Research interests - hydroelasticity, non-linear problems of bending of beams and plates made of materials with the SD effect, the effect of ice on the strength of hydraulic structures supports

### March 1, 2022

I. K. Patel (SPbSU)

CONTACTLESS CAPTURE AND REMOVAL USING ELECTROMAGNETIC INDUCTION

A new method of contactless removal of space debris using electromagnetic
induction is considered. The possibility to clean near-Earth space off of metallic
space debris objects with the help of magnetically assisted spacecraft (also called
collector) is considered. The proposed active debris removal method is classified as
contactless. In order to capture a heavy satellite from high Earth orbit and
subsequently, transferring it to a target point in low Earth orbit where it can be safely
injected into the Earth’s atmosphere its motion should be accurately predicted. In
addition, it is necessary for the success of any mission to optimize the fuel
consumption in the presence of gravitational perturbation. Therefore, the need to
consider the influence of perturbing forces such as Earth’s oblateness is necessary.
In this model, the perturbation due to J_{2} zonal harmonics is taken into account. The orbital dynamics of space debris is studied using numerical simulations. The applicability of the idea is discussed based on simulations, and strategies for
improvement and further study are considered.

**
Patel Ishan Kirankumar ** – PhD student at St. Petersburg State University,
Department of Theoretical and Applied Mechanics, Faculty of Mathematics and Mechanics.
Research interests: Methods of active debris removal. Scientific adviser – Prof. A.A. Tikhonov

### February 22, 2022

A.H. Gabrielyan (SPbSU)

Peculiarities of deformational behavior of shape memory alloys

The problem of deformational behavior of shape memory alloys is discussed In the report. The research intends to study the effect of pseudoelasticity during isothermal holding under pressure, as well as the effect of martensite stabilization in quenched Ti49Ni51 alloy samples. A number of experimental data has been obtained on strain variation and temperatures transition in the studied alloy. It is shown that strain variation increases during isothermal holding and the effect of strain growth reaches a plato with cycles. The hypothesis about the influence of boundary damage on the effect of martensite stabilization was put forward.

**
Artur H. Gabrielyan ** – master's student at St. Petersburg State University, Department of Elasticity Theory, Faculty of Mathematics and Mechanics. Research interests: deformation behavior of shape memory alloys. Scientific adviser – Doctor of Physics and Mathematics S.P. Belyaev.

### February 15, 2022

Degilevich E. A., Smirnov A. S. (SPbPU)

The modeling and optimization of the catenary and its modifications

The report discusses the modeling and optimization of a catenary and its several modifications, namely: an inextensible catenary in a homogeneous gravitational field, an extensible catenary in a homogeneous gravitational field, and also an inextensible catenary in a Newtonian gravitational field. An exact analytical solution is given for the catenary models under consideration, during which the optimal values of the length of the catenary are determined, ensuring minimum tension forces on the suspension points. Creation of discrete models with concentrated parameters is carried out in the ADAMS software package. The results of numerical experiments are compared with analytical solutions for continuous catenary models with distributed parameters. As a result of the study, graphs of the force characteristic are constructed depending on the length of the catenary, on the basis of which the correctness of discrete models is confirmed.

**
Degilevich Egor A. ** – Second year graduate student of the Higher school of mechanics and control processes, Peter the Great St. Petersburg Polytechnic University. Research interests: analytical mechanics, theory of oscillations, stability theory, optimization in mechanics.

**
Smirnov Alexey S. ** – Assistant Professor of the Higher school of mechanics and control processes, Peter the Great St. Petersburg Polytechnic University. Research interests: analytical mechanics, theory of oscillations, stability theory, optimization in mechanics.

### November 23, 2021

S.N. Burian (SPbSU)

Dynamics of mechanisms with singularities

Configuration space of mechanical system in holonomic mechanics is a smooth manifold. Smooth structure gets us a possibility to define motion equations and corresponding vector field on the phase space of the system. In the case that there are singular points in the configuration space, only particular methods are applied. We consider several theories of singular space geometries which generalize base terms of differential calculus: (co)tangent vector, (co)tangent space and vector field, integral curves of vector fields, etc. In order to study the application of these theories to the problems of analytical dynamics, some particular examples of mechanical systems with singularities are considered. Configuration spaces, smooth motion types and reaction forces are studied. Dynamics, which is constructed in the frames of geometric theories, is compared with the observed dynamics of model examples of mechanisms. This comparison could help up to formulate the conditions which generalized geometric theory of motion must satisfy.

**
Sergey N. Burian ** – PhD student of the Department of Higher Geometry of Faculty of Mathematics and Mechanics at St. Petersburg State University. Research area – analytical mechanics, geometry of singular spaces. Scientific supervisor – Asso. Prof. V.S. Kalnitsky.

### November 9, 2021

M.A. Bushmakova (SPbSU)

Evaluation of relaxation terms in state-to-state approachment problems using machine learning methods

Relaxation terms characterize the variation of vibrational level populations of molecules and molar fractions of atoms as a result of various types of energy exchange and chemical reactions. They are the right-hand sides of the set of differential equations for macroscopic gas parameters, describing the flows of multicomponent reacting gas mixtures under conditions of strong vibrational and chemical non-equilibrium. Conventional methods for calculating relaxation terms are computationally expensive, since they imply multiple summations, as well as the calculation of a large number of the rate coefficients of vibrational energy transitions and chemical reactions. In this work, a possibility of evaluating the relaxation terms by machine learning methods is assessed, and the efficiency of two models learned on datasets calculated with SSH and FHO methods is compared for the case of VT and VV relaxation in the mixture of O_{2}-O. Then the solution of zero-dimensional problem obtained by ML algorithms is compared to solution with SSH and FHO methods. It is shown that the k-nearest neighbours algorithm provides the best accuracy/efficiency ratio and can be recommended for further studies. It is also possible to decrease computational time by using ML algorithms.

**
Mariia A. Bushmakova ** – first year master student of the Department of Hydroaeromechanics at St. Petersburg State University. Project supervisor - Prof. E.V. Kustova.

### October 12, 2021

E.P. Nosov (SPbSU)

Study of the diffraction problem by the ray method for reflection of a plane wave from a parabolic surface

One of the methods for analysis of the diffraction problem is the ray method, the current concept of which was formed at the end of the last century. Nowadays, there is no general formula describing the reflection of a wave from an arbitrary surface. Therefore, special cases are considered. An example of such a case is the problem of external reflection from a parabolic surface. The purpose of this paper is a detailed study of that problem. In the current research the first two terms of the formal asymptotic series for the reflected plane monochromatic wave with constant velocity were found; the asymptotics and the residual were also calculated. It appears that the first term of the formal expansion gives sufficient accuracy for a number of problems.

**
Evgeny P. Nosov ** – first year master student of the Department of theoretical and applied mechanics at St. Petersburg State University. Project supervisor - Prof. I.V. Andronov.

### October 5, 2021

S.O. Bondarenko (SPbSU)

Motion of a horizontal dynamically unbalanced rotor in gravitational field.

A horizontal dynamically unbalanced rotor motion is considered. The motion equations of two systems are obtained: with two-plane automatic ball balancer and without it. Two rotor models are designed in MATLAB Simulink base on simplyfied equation systems for numerical solution. The whirling amplitude equations are obtained analytically for a rotor rotating at constant angular speed. The gravity is taken into account. Numerical and analytical results are compared.

**
Sergei O. Bondarenko ** – first year master student of the Department of theoretical and applied mechanics at St. Petersburg State University. Research area - rotordynamics. Scientific supervisior - Asso.Prof. A. S. Kovachev.