ON STABILIZATION OF UNSTABLE ROTATION IN THE RESISTING MEDIUM OF THE LAGRANGE GYROSCOPE USING THE SECOND GYROSCOPE AND ELASTIC HINGES

The possibility of stabilizing an unstable uniform rotation in a resisting medium of a "sleeping" Lagrange gyroscope using a rotating second gyroscope and elastic spherical hinges is considered. The "sleeping" gyroscope rotates around a fixed point with an elastic recovery spherical hinge, and the second gyroscope is located above it. The gyroscopes are also connected by an elastic spherical restorative hinge and their rotation is supported by constant moments directed along their axes of rotation. It is shown that stabilization will be impossible in the absence of elasticity in the common joint and the coincidence of the center of mass of the second gyroscope with its center. With the help of the kinetic moment of the second gyroscope and the elasticity coefficients of the hinges, on the basis of an alternative approach, the stabilization conditions obtained in the form of a system of three inequalities and the conditions found on the elasticity coefficients at which the leading coefficients of these inequalities are positive. It is shown that stabilization will always be possible at a sufficiently large angular velocity of rotation of the second gyroscope under the assumption that the center of mass of the second gyroscope and the mechanical system are below the fixed point. The possibility of stabilizing the unstable uniform rotation of the "sleeping" Lagrange gyroscope using the second gyroscope and elastic spherical joints in the absence of dissipation is also considered. The "sleeping" gyroscope rotates at an angular velocity that does not meet the Mayevsky criterion. It is shown that stabilization will be impossible in the absence of elasticity in the common joint and the coincidence of the center of mass of the second gyroscope with its center. On the basis of the innovation approach, stabilization conditions were obtained in the form of a system of three irregularities using the kinetic moment of the second gyroscope and the elastic coefficients of the hinges. The condition for the angular momentum of the first gyroscope and the elastic coefficients at which the leading coefficients of these inequalities are positive are found. It is shown that if the condition for the angular momentum of the first gyroscope is fulfilled, stabilization will always be possible at a sufficiently large angular velocity of rotation of the second gyroscope, and in this case the center of mass of the second gyroscope can be located above the fixed point.

SPATIAL PROBLEMS OF DYNAMIC STABILITY OF FRAME STRUCTURES

Periodic longitudinal forces in structural elements caused by operational or seismic influences, at certain values of the parameters of these forces can cause the occurrence and growing of transverse oscillations of these elements. This phenomenon is called parametric resonance or loss of dynamic stability. In the works of N. M. Belyaev, N. M. Krylov, М. М. Bogolyubov, E. Mettler, V. N. Chelomey, V. V. Bolotin flat problems of dynamic stability of frame structures were investigated. In this paper the modified Bolotin’s method, proposed to solve flat problems of dynamic stability of frames, is used. Instead of the deformation method used by V. V. Bolotin to construct analytical expressions of deflections of frame rods, in the modified method the numerical-analytical method of boundary elements is used. The article proposes a method for constructing domains of dynamic instability of frames in the space of parameters (frequency and amplitude) of seismic and operational dynamic influences that cause longitudinal forces in the rods, which periodically change over time and lead to unlimited growth of transverse oscillations amplitudes in the domains of instability. The proposed method is demonstrated in example, which considers the spatial problem of dynamic stability of a П-shaped frame with two concentrated masses located on it, which are under the action of vertical periodic forces. These forces create periodic longitudinal forces in the vertical rods of the frame. Areas of dynamic instability of the frame were constructed. Taking into account the destructive effect of oscillations is important for practical application. The most dangerous destructive effect of oscillations is observed in earthquakes and explosions. The study of this action makes it possible to avoid undesirable consequences of oscillations by limiting their level and to solve important practical problems of the dynamics of structures. Solving dynamics problems is a difficult problem. Dynamic calculation of structures provides their bearing capacity under the combined action of static and dynamic loads.

EIGENFREQUENCIES OF OSCILLATIONS OF A PLATE WHICH SEPARATES A TWO-LAYER IDEAL FLUID WITH A FREE SURFACE IN A RECTANGULAR CHANNEL

The frequency spectrum of plane vibrations of an elastic plate separating a two-layer ideal fluid with a free surface in a rectangular channel is investigated analytically and numerically. For an arbitrary fixing of the contours of a rectangular plate, it is shown that the frequency spectrum of the problem under consideration consists of two sets of frequencies describing the vibrations of the free surface of the liquid and the elastic plate. The equations of coupled vibrations of the plate and the fluid are presented using a system of integro-differential equations with the boundary conditions for fixing the contours of the plate and the condition for the conservation of the volume of the fluid. When solving a boundary value problem for eigenvalues, the shape of the plate deflection is represented by the sum of the fundamental solutions of a homogeneous equation for a loose plate and a partial solution of an inhomogeneous equation by expanding in terms of eigenfunctions of oscillations of an ideal fluid in a rectangular channel. The frequency equation of free compatible vibrations of a plate and a liquid is obtained in the form of a fourth-order determinant. In the case of a clamped plate, its simplification is made and detailed numerical studies of the first and second sets of frequencies from the main mechanical parameters of the system are carried out. A weak interaction of plate vibrations on vibrations of the free surface and vice versa is noted. It is shown that with a decrease in the mass of the plate, the frequencies of the second set increase and take the greatest value for inertialess plates or membranes. A decrease in the frequencies of the second set occurs with an increase in the filling depth of the upper liquid or a decrease in the filling depth of the lower liquid. Taking into account two terms of the series in the frequency equation, approximate formulas for the second set of frequencies are obtained and their efficiency is shown. With an increase in the number of terms in the series of the frequency equation, the previous roots of the first and second sets are refined and new ones appear.

SOME PROBLEMS OF OPTIMIZATION AND CONTROL OF THE NATURAL FREQUENCIES OF AN ELASTICALLY SUPPORTED RIGID BODY

The article analytically investigates the behavior of the frequencies and modes of natural vibrations of a rigid body, based on point elastic supports, when the position of the supports changes. It is assumed that the body is in plane motion and has two degrees of freedom. A linear description of body vibrations is accepted. The problems of determining such optimal positions of elastic supports at which the fundamental frequency of the structure reaches its maximum value are considered. Two groups of problems were studied. The first group concerns a body supported by only two supports. It was found that in the absence of restrictions on the position of the supports to maximize the fundamental natural frequency, these supports should be positioned so that the basic natural vibrations of the body are translational. Simple analytical conditions are formulated that must be satisfied by the corresponding positions of the supports. In real practical situations, these positions may be unreachable due to the presence of various kinds of restrictions due to design requirements. In this paper, optimization problems are considered taking into account a number of restrictions on the position of supports, typical for practice, expressed analytically by equations and inequalities. For each of the considered types of constraints, results are obtained that determine the optimal positions of the supports and the corresponding maximum values of the main natural frequencies. The approach applied allows us to consider other types of restrictions, which are not considered in the article. In the second group of problems for a body resting on an arbitrary number of supports, the optimal position of an additional elastic support introduced in order to maximize the fundamental frequency in fixed positions and the stiffness coefficients of the remaining supports was sought. It was found that this position depends on the value of the stiffness coefficient of the introduced support. Results are obtained that qualitatively and quantitatively characterize this position and the corresponding frequencies and modes of natural oscillations, including taking into account practically established limitations. The research method uses a qualitative approach, systematically based on the well-known Rayleigh theorem on the effect of imposing constraints on the free vibrations of an elastic structure.

EXPERIMENTAL STUDIES OF DEFORMABILITY AND FRACTURE RESISTANCE OF AIRFIELD SLABS ON MODELS

The results of experimental studies of deformability and crack resistance of models of aerodrome slabs made of reinforced concrete and steel-fiber concrete, made on the basis of serial slab PAG-18 taking into account the scale factor, are presented. Two series of slabs were tested - two models of reinforced concrete and two models with one-percent dispersed reinforcement. The load was applied in steps, the instrument readings were recorded twice at each step and the crack opening width was measured starting from the moment of the first crack formation. Dial gauges, deflectometer and microscope MPB-3 were used as measuring instruments. In accordance with the normative documents acting in Ukraine, one of two possible loading schemes was considered - with the loading by the concentrated force applied in the span part of a plate which had a hinged support along its short sides. Plate models were tested on a specially made stand. Each load step ended with a five-minute dwell time, at the beginning and the end of which readings were taken on the measuring instruments. The deformations at the same levels were measured with dial gauges. The process of crack formation was observed with a Brinell tube in the places of the greatest crack opening. Breaking load for fiber concrete slab was 1.52 times higher than for reinforced concrete slab, and the moment of cracking initiation was 1.22 times higher. The process of cracking in the fiber concrete slab begins at higher loads than in the reinforced concrete slab. The initial crack opening width of the slabs is almost the same, and the final crack opening width of all the cracks in the fiber concrete slab is significantly lower than in the reinforced concrete slab. The deformations in steel-fiber concrete slabs when the load is applied in the span, both for compressed and stretched fibers, are higher than in reinforced concrete slabs. The experimental studies indicate that dispersed reinforcement of airfield slabs with steel fiber leads to their higher crack resistance.

EVALUATION OF FIRE RESISTANCE OF FIRE PROTECTED STEEL STRUCTURES BY CALCULATION AND EXPERIMENTAL METHOD

Based on the developed geometric, physical, computer and finite element model, the fire resistance of fire-resistant steel structures was evaluated by calculation and experimental method. The adequacy of the developed computational-experimental method for assessing the fire resistance of fire-resistant steel structures in assessing the fire resistance of a fire-resistant I-beam steel column was verified. The results of tests for fire resistance of steel columns with fire-retardant coating at standard temperature of the fire without the load applied to them (temperature in the furnace, temperature in certain places on the surface of fire-retardant steel columns, the behavior of the investigated fire-retardant coating). The analysis of tests on fire resistance of fire-resistant steel columns exposed to fire at standard temperature (temperature in the furnace, temperature in places of measurement of temperature on a surface of columns, behavior of a fire-retardant covering) is carried out. A computer model of the «steel column – reactive flame retardant coating» system has been built for numerical simulation of non-stationary heating of such a system. Simulation of non-stationary heating of the system «steel column – fire-retardant coating» in the software package FRIEND with the specified parameters (geometric model, thermal effects, initial and boundary conditions, properties of system materials). The reliability of the results of numerical modeling with real experimental data on the duration of fire exposure at the standard temperature of the fire to reach the critical temperature of steel. Based on the comparison of experimental results and numerical simulations, a conclusion is made about the adequacy of the developed model to the real processes that occur when heating fire-retardant steel columns without applying a load under fire conditions at standard fire temperature. The efficiency of the proposed calculation and experimental method for assessing the fire resistance of fire-resistant steel structures has been confirmed.

FORMATION OF MATHEMATICAL APPARATUS OF METHODS OF FIRE AND EXPLOSION SAFETY CONTROL OF LANDFILL

The authors analyze the fire and explosion hazards of solid waste disposal facilities, taking into account current trends in the introduction of biogas (methane) collection and utilization systems. Methane is considered an alternative energy source for power plants. The authors determined the initial and limiting conditions of the mathematical apparatus of the method of combating fire and explosion hazards of solid waste disposal facilities based on the results of analysis and synthesis of factors of occurrence and spread of man-caused danger, existing mathematical models, and methods of counteracting man-caused danger. This is the basis for the further development of appropriate emergency response techniques. During the analysis, the authors found that humidity, the temperature of the landfill (household waste), the presence of sufficient oxygen at some point in time initiate the formation of explosive concentrations of methane in the array and contribute to the spread of hazards in landfills or dumps. The specific weight of the organic component, the value of the density of the array, the height of the landfill affect the process of counteracting the danger, namely the prevention of dangerous events and prevention of emergency from the object to the highest level of distribution (local level), primarily in the first group priorities, such as the number of victims and injured civilians and specialists of the units of the State Emergency Service of Ukraine. The team of authors has defined a system of equations of connection of the existence of the mathematical device taking into account initial and boundary conditions. A system of communication equations is determined taking into account the initial and boundary conditions of the mathematical apparatus, which allows to further develop a control algorithm for emergency response related to fire and explosion hazardous landfills close to settlements.

CUTTING STONE BUILDING MATERIALS WITH CUT WHEELS OF CUBIC BORON NITRIDE

The advantage of cutting stone building materials with SSM (synthetic superhard materials) wheels is that, first of all, it is possible to obtain high processing productivity and dimensional stability, which are 3..5 and 50…100 times higher than those of traditional tools based on carborundum, respectively. The study of the process of cutting stone materials with CBN (cubic boron nitrite) wheels is aimed at establishing force dependences, determining the cutting power and heating temperature of the cutting disc during operation. The forces were measured using a tensometric dynamometer UDM-50. To measure and calculate the cutting temperature, a thermoelectric method based on the formation of practically not inertial microthermocouples during cutting was used. The temperature to which the CBN cutting wheel on a metal base is heated is a limiting factor in processing, since when heated to a temperature of 600ºС, the strength of the wheel decreases by half, which can cause its rupture under the action of centrifugal forces, as well as loss of stability and jamming during cutting. In the present study, the wheel temperature was measured after one minute of continuous operation. The values of the component of the cutting force PY, depending on the processing modes, can reach values of the order of 70 N. The values of the component of the cutting force PZ, depending on the processing modes, can reach values of 45 N. The cutting power can be 2800W. The temperature resistance of the wheel (heating time of the wheel up to 600ºС) when cutting dry is maximum 28 minutes, when grinding with cooling of the cutting zone with negative temperature air from a Ranque-Hilsch tube, the temperature resistance is 35 minutes, with ejector cooling of sprayed coolant 37 minutes and with jet-pressure cooling it is 40 minutes. The maximum cutting length is respectively 0.7: 0.8; 0.9 and 2m. The cutting power is 600...2800W.

SOME PROBLEMS ABOUT THE MOTION OF A RIGID BODY IN A RESISTIVE MEDIUM

The dynamics of rotating rigid bodies is a classical topic of study in mechanics. In the eighteenth and nineteenth centuries, several aspects of a rotating rigid body motion were studied by famous mathematicians as Euler, Jacobi, Poinsot, Lagrange, and Kovalevskya. However, the study of the dynamics of rotating bodies of still important for aplications such as the dynamics of satellite-gyrostat, spacecraft, re-entry vehicles, theory of gyroscopes, modern technology, navigation, space engineering and many other areas. A number of studies are devoted to the dynamics of a rigid body in a resistive medium. The presence of the velocity of proper rotation of the rigid body leads to the apearance of dissipative torques causing the braking of the body rotation. These torques depend on the properties of resistant medium in which the rigid body motions occur, on the body shape, on the properties of the surface of the rigid body and the distribution of mass in the body and on the characters of the rigid body motion. Therefore, the dependence of the resistant torque on the orientation of the rigid body and its angular velocity can de quite complicated and requires consideration of the motion of the medium around the body in the general case. We confine ourselves in this paper to some simple relations that can qualitative describe the resistance to rigid body rotation at small angular velocities and are used in the literature. In setting up the equations of motion of a rigid body moving in viscous medium, we need to consider the nature of the resisting force generated by the motion of the rigid body. The evolution of rotations of a rigid body influenced by dissipative disturbing torques were studied in many papers and books. The problems of motion of a rigid body about fixed point in a resistive medium described by nonlinear dynamic Euler equations. An analytical solution of the problem when the torques of external resistance forces are proportional to the corresponding projections of the angular velocity of the rigid body is obtain in several works. The dependence of the dissipative torque of the resistant forces on the angular velocity vector of rotation of the rigid body is assumed to be linear. We consider dynamics of a rigid body with arbitrary moments of inertia subjected to external torques include small dissipative torques.

MODAL ANALYSIS OF REINFORCED CONCRETE AND FIBER CONCRETE BEAMS

Analytical, experimental and numerical results of determination of natural frequencies and forms of oscillations of reinforced concrete and fiber concrete beams are given. Modern analytical, numerical and experimental methods of studying the dynamics of reinforced concrete and fiber concrete beams are analyzed. The problem of determining the natural frequencies and forms of oscillations of reinforced concrete and fiber concrete beams at the initial modulus of elasticity and taking into account the nonlinear diagram of deformation of materials is solved analytically. Computer modeling of the considered constructions in four software complexes is done and the technique of their modal analysis on the basis of the finite element method is developed. Experimental researches of free oscillations of the considered designs and the comparative analysis of all received results are carried out. It is established that all involved complexes determine the imaginary frequency and imaginary form of oscillations. The frequency spectrum calculated by the finite element method is approximately 4% lower than that calculated analytically; the results of the calculation in SOFiSTiK differ by 2% from the results obtained in the PC LIRA; the discrepancy with the experimental data reaches 20%, and all frequencies calculated experimentally, greater than the frequencies calculated analytically or by the finite element method. This rather significant discrepancy is explained, according to the authors, by the incorrectness of the used dynamic model of the reinforced beam. The classical dynamics of structures is known to be based on the theory of linear differential equations, and the oscillations of structures are considered in relation to the unstressed initial state. It is obvious that in the study of free and forced oscillations of reinforced concrete building structures such an approach is unsuitable because they are physically nonlinear systems. The concept of determining the nonlinear terms of these equations is practically not studied. Numerous experimental researches and computer modeling for the purpose of qualitative and quantitative detection of all factors influencing a spectrum of natural frequencies of fluctuations are necessary here.