Індекс рубрикатора НБУВ: В161.61

Рубрики:

Звичайні диференціальні рівняння

Шифр НБУВ: Ж14648 Пошук видання у каталогах НБУВ 

Індекс рубрикатора НБУВ: В2в51

Шифр НБУВ: Ж22412/а Пошук видання у каталогах НБУВ 

Model of Earth charge resonant oscillations excitations based on Tesla experiment, was offered. Solutions of d'Alembert wave equations for electric and magnetic potentials of the charged perfectly conductive sphere were found. Graphic analyses of perturbed potential distribution on the Earth surface was provided. It was shown that obtained solution provides adequate description of Tesla experiment on wireless transfer of electricity conducted in Colorado Springs in 1899 - 1900. In the very low-frequency area of the electro-magnetic oscillations spectrum it was offered to consider the Earth as capacitors battery, consisting of two put one into another perfectly conductive spheres, between which thin dielectric layers is placed. Own oscillation frequencies of the Earth charge were determined in this area. It was shown that these frequencies most precisely correspond to experimentally measured Schumann resonances.

Запропоновано огляд попередніх і нових досліджень класичної проблеми М. А. Буля опису узгоджених лінійних векторних полів та її загальних розв'язків М. Г. Пфайфера, а також нових спеціальних розв'язків тину Лакса - Сато. Проаналізовано відповідні алгебричні структури Лі та властивості інтегровності для дуже цікавого класу нелінійних динамічних систем, що називаються бездисперсійними рівняннями небесного типу, які були введені Є. Плебанським і пізніше проаналізовані в серії статей. На базі Лі-алгебричної схеми Адлера - Костанта - Суріо та асоційованої з нею R-структури вивчено орбіти відповідних коприєднаних дій, тісно пов'язаних із класичними структурами Лі - Пуассона на них. Показано, що умова сумісності для них збігається з відповідними рівняннями небесного типу, що аналізуються. Показано, що всі ці рівняння породжуються таким чином і їх можна представити як умову сумісності Лакса для спеціально побудованих петельних векторних полів на торі. Описано нескінченну ієрархію законів збереження, пов'язаних із небесними рівняннями, та вказано на її аналітичну структуру, що пов'язана з інваріантами Казіміра. Наведено типові приклади таких рівнянь, для яких проаналізовано їх інтегровкість у межах запропонованого підходу. Показано зв'язок розвиненого підходу з дуже цікавою класичною механічною інтерпретацією Лагранжа - Даламбера.

Purpose. Research on interrelation of the efficiency of non-isometric particle destruction due to flexural deformation with the parameters of operational parts of new designs of roll and vibrational centrifugal disintegrators. Methodology. A mathematical model of quasistatic flexural deformations of ellipsoid-shaped particles is developed for the case of their nip between the wave-profiled rolls of disintegrator, based on the classical problem of a beam bend with longitudinal compression. A mathematical model of inertial destruction of ellipsoid-shaped particles during free impact in a vibrational two-shaft centrifugal module is created based on combination of the beam bend problem, the contact deformations theory of Hertz and d'Alembert principle. The dependences of the key efficiency parameters of the mentioned disintegrators on the geometrical parameters of particles of processed mining rocks are obtained by the methods of generalization and dimensionless parametrization. Findings. The analytical model of flexural tension realization in ellipsoid-shaped particles for disintegrator having the wave profile of rolls has allowed establishing that the increase in the length coefficient of ellipsoid-shaped particles from 1,5 to 4 units leads to the enhancement of breaking stresses from 1m 7 to 12 times, compared to the case of smooth rolls. The analysis of the model of inertial disintegration of ellipsoid-shaped particles has revealed that the destruction of particles' narrow fractions in counter flows saves up to 20 % of energy in comparison to the destruction by the rigid barrier. The particles of less than average size are destructed the most efficiently, during processing in counter flows and being in fractions of unequigranular structure. But the disintegration becomes more difficult as the relative size of particle rises. Originality. Two mathematical models, which take into consideration the influence of flexural deformations of non-isometric particles modelled by ellipsoids on the level of breaking stresses for the disintegrators with wave profile of rolls, and also on the minimal speed of particles inertial flow for the vibrational two-shaft centrifugal module, are developed and analyzed. Practical value. The obtained results allow determining the key parameters of operational parts for new designs of disintegrators. This forms the basis for development of techniques for calculation of operational parts of modern samples of crushing and grinding equipment.

Problem statement. Mobile robotic systems are widely used in various fields of industry and social life: from small household appliances to large-size road-building machinery. Specific attention of scientists and designers is paid to the vibration-driven locomotion systems able to move in the environments where the use of classical wheeled and caterpillar robots is impossible or inefficient. Purpose. The main objective of this paper consists in generalizing the actual research results dedicated to various design diagrams and mathematical models of suspension systems of mobile vibration-driven robots. Methodology. The differential equations describing the robot motion are derived using the Lagrange - d'Alembert principle. The numerical modeling is carried out in the Mathematica software by solving the derived system of differential equations with the help of the Runge - Kutta methods. The verification of the obtained results is performed by computer simulation of the robot motion in the SolidWorks and MapleSim software. Findings (results). The time dependencies of the basic kinematic parameters (displacement, velocity, acceleration) of the robot's vibratory system are analyzed. The possibilities of maximizing the robot translational velocity are considered. Originality (novelty). The paper generalizes the existent designs and mathematical models of the mobile vibration-driven robots' suspensions and studies the combined four-spring locomotion system moving along a rough horizontal surface. Practical value. The obtained results can be effectively used by researchers and designers of vibration-driven locomotion systems while improving the existent designs and developing the new ones. Scopes of further investigations. While carrying out further investigations on the subject of the paper, it is necessary to solve the problem of optimizing the robot's oscillatory system parameters in order to maximize its translational velocity.