Sessions 2022-2023

The report addresses a problem related to ballistics, a field that studies the motion of objects in the gravitational field of Earth. Ballistics involves calculating trajectories, determining velocities and accelerations, and accounting for various factors that affect motion, such as the Coriolis effect and medium resistance. The effect of Coriolis force on ballistic trajectories and optimal parameters is studied without taking into account other factors. The solution to the problem of projecting a body at an angle with respect to the horizontal plane has been explicitly derived. Due to the low angular velocity of Earth's rotation, a complicated solution was significantly simplified by breaking it down into a small parameter and omitting insignificant terms.

For the task of projecting at a given point, the main optimization criteria considered are energy and energy-time efficiency. Using these criteria, optimal initial velocities and angles of projection to the horizon can be determined.

The moment of inertia of the solid and hollow regular n-gon cross-section with respect to the axis passing through its center in the section plane is evaluated. The ability to take such points into account is necessary when analyzing the deformations, stability and vibrations of beams and prismatic shells. The method discussed in the paper allows one to analytically calculate axial moments of inertia for regular polygonal sections of beams and shells with an arbitrary number of sides.

The accurate calculation of transport coefficients in high-speed flow modeling of reacting air is essential in the analysis of heat transfer processes in various problems of gas dynamics, for example, in design of thermal protection of descent vehicles. The most accurate from the physical point of view within the continuum approximation is the state-to-state approach. However, it has a significant computational complexity. One of the most effective and accessible method of applying such models is a machine learning regression. We study the application of this approach to the problem of modeling viscosity and thermal conductivity coefficients. Further prospects of machine methods in modeling of high-speed flows are considered.

The report is devoted to methods for measuring ultra-low mass of deposited particles using micromechanical sensors based on the phenomenon of mode localization. Two types of sensitive elements of a mass detector are considered – a beam with an initial curvature and a system of two mechanically weakly coupled beams. In the first case, mass detection is supposed to be carried out by the amplitude ratio of beam vibrations on different natural modes. It is shown that with the correct choice of geometric parameters in the system, an exchange of energy is observed between the first asymmetric and second symmetric modes of vibration. The work proposes a system of electrodes that makes it possible to excite and detect vibrations in selected forms. In the case of a system of two weakly coupled beams, it is shown that in the presence of a disturbance in the form of a deposited particle, an oscillation mode with different amplitudes is observed in the system, the ratio of which is the output signal of the device. A comparison is made of the output characteristics of the devices, primarily sensitivity, with known mass detectors presented in the literature.

The report considers nonlinear oscillations of a double mathematical pendulum with identical parameters of links and end weights. Exact nonlinear equations of motion of the system are derived, from which a classical linear model of small oscillations is obtained, as well as a weakly nonlinear oscillation model that takes into account cubic nonlinearity. A well-known solution of the problem of small oscillations of a double pendulum is given, which serves as a basis for further research. With the help of asymptotic methods of nonlinear mechanics, an approximate solution is constructed for a weakly nonlinear model under arbitrary initial conditions of motion. It is shown that the found solution has a nontrivial structure and represents polyharmonic oscillations on eight different frequencies. The resulting approximate formulas are accompanied by graphical illustrations that compare the behavior of the angles of deviation of the links of a double pendulum from the vertical when using linear and weakly nonlinear models, as well as the original nonlinear model, which is calculated using numerical integration. The conclusions drawn are of interest for analytical mechanics and theory of oscillations, and they can also be used in practice in applied problems of robotics and biomechatronics.