Sessions 2024-2025

The report addresses the problem of hydrostatics: the bending of a thin ice plate resting on an elastic foundation under a concentrated vertical load, which simulates the landing and takeoff of helicopters. The study includes the calculation of the minimum ice thickness required for safe landing, as well as an analysis of the stress-strain state (SSS) of the ice. The ice is treated as a floating plate or beam. This problem is analogous to the classical Hertz problem—the bending of a thin circular plate subjected to a vertical load applied at the center of its upper surface.

The bending of the ice plate is examined in two configurations. In the first configuration, the plate is rigidly clamped along its edges. In the second configuration, the plate is also clamped along its edges but rests on an elastic foundation, modeled as the water beneath the ice. Mathematical modeling and calculations are performed using the finite element method in the ANSYS 15.0 software for two loading scenarios: first, the plate is loaded at its center, and then the load is applied at three points, corresponding to the actual conditions of a helicopter landing on ice.

Two models are compared: a phenomenological model based on experimental data and a mathematical model implemented in ANSYS using the finite element method. The results demonstrate a qualitative agreement.

The report is devoted to the problems of development and verification of computational algorithms for the synthesis of compact models of the dynamics of continual elastic systems in a geometrically nonlinear formulation (primarily thin-walled structures: strings, membranes, beams, plates, shells) based on the finite element method. The approaches under consideration are based on the idea of??identifying a nonlinear (quadratic-cubic) stiffness characteristic of an elastic system in its modal coordinates with the subsequent application of the apparatus of the theory of nonlinear normal modes and Poincare normal forms to construct an invariant manifold tangent to the modal subspace of interest. The resulting dynamic reduced-order model takes into account the nonlinear elastic coupling of the working vibration modes with high-frequency longitudinal and volume modes of the structure, which ensures the correctness of the calculated nonlinear stiffness characteristic of the system for the selected principal coordinates. The developed algorithm is applied to a number of problems of nonlinear dynamics of strings and beams that admit an approximate analytical solution using asymptotic methods of nonlinear mechanics. The features of the software implementation of the presented method based on the ABAQUS finite element analysis software system are discussed.

Well engineered coaxial airscrew offers advantage in efficiency to airplane’s propulsion system. However modern approach on designing coaxial propellers is based on analysing and semi-empirical enhancing of already existing geometries. Present study provides realised in code and validated method for computing thrust and power coefficients for low and medium loaded airplane’s coaxial airscrew.

New approach is based on combination of elder approaches on aerodynamic calculation of coaxial airscrew and on aerodynamic calculation of highly loaded airscrew. The idea is then realized with implementation of convenient computational schemes and validated through comparison with full-scale experiment. Close agreement of calculations and experiment is found in areas of low and medium loads.

Present study is used to demonstrate the potential of such approach, as it only points out the way and necessity of further research.

Problems that arise when constructing mechanical models of the human body subject to vibration are discussed. In particular, the previously unconsidered question of the uniqueness of the set of parameters of such models is examined, the questions that is fundamentally important when using them in the construction of vibration protection systems.

An example of the simplest mechanical model of a chain structure located on a vibrating base and consisting of two masses connected in series by linear springs and dampers is considered. The necessary conditions for the uniqueness of the model parameters are found. Next, a study is carried out of systems simulating the muscular structure of the human body (with multi-link connections) and the intervertebral disc (with a non-integer number of degrees of freedom). Computer modeling made it possible to identify the features of such models.

The work is devoted to the experimental study of oscillations of a parallelepiped in an air stream. Poorly streamlined bodies of a similar shape, serving as parts of bridges or high-rise structures, are capable of being subjected to fluctuations of various types in the incoming wind flow.

In the study, a parallelepiped with end washers was fixed in the working part of the wind tunnel on two types of suspensions containing springs. The body on the suspension containing eight springs had only steady translational oscillations in the direction perpendicular to the velocity of the incoming flow. For a body on a suspension containing two springs, at sufficiently high flow velocities, only rotational oscillations with a constant amplitude were observed. Although the parallelepiped can perform both translational and rotational vibrations under the action of the airflow, the occurrence of both translational and rotational vibrations on the same suspension was not observed.

Using the methods of tensometric measurements of unsteady forces in an aerodynamic experiment, after processing the data, it was found that the square of the oscillation amplitude linearly depends on the Struhal number.

The electrocoalescence of droplets is studied - the process of merging liquid droplets under the effect of an electric field. This process plays an important role in a variety of process applications including liquid handling, emulsion separation and fuel purification.

During the study, a map of droplet interaction modes was created, showing how the form of droplet fusion depends on the electric field strength and the droplet size ratio.

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.