Natural Oscillation Frequency Research Articles

The natural vibrations of shell structures taking into account dissipative properties and structural heterogeneity

The paper considers the natural oscillations of shell structures. In general, these structures are a set of deformable elements with different rheological properties. An algorithm for solving the viscoelastic dynamic problems has been developed for the complex axisymmetric structures. The physical properties of the viscoelastic structural elements are described by linear Boltzmann-Voltaire relations with integral difference cores. The three-parameter core of Rzhanitsyn-Koltunov was used as the relaxation core. In general, the problem is reduced to solving the systems of first-order ordinary differential equations in the complex variables. A frequency equation is obtained, for the solution where the Muller method is applied. The calculated values of the natural frequencies of oscillations with a given degree of accuracy are given.

TESTING OF BRIDGES DAMAGED AS A RESULT OF COMBAT ACTIONS

The work examines the application of methods of static and dynamic testing of bridges damaged as a result of hostilities. Penetration by projectiles and damage by explosions of structural elements, in addition to obvious damage, also form a series of hidden defects, the determination of which according to norms and regulatory documents is regulated to be carried out by means of tests with calculated loads. For testing bridges in accordance with the requirements of construction standards and regulatory documents, the equipment developed by the author of the article was used for digital registration and processing of signals of acoustic emission and mechanical vibrations. The results of static tests of three span overpasses made of precast concrete with a length of 63.20 m along the street are presented. Kvitneva in the city of Chernihiv and a four-span pedestrian bridge with a length of 263.1 m across the Desna River in the park zone of the city of Chernihiv. The peculiarities of the application of the acoustic emission method for determining the state of the span structures of the bridge during static tests with step loading are considered. During the analysis of the results of the dynamic tests of the pedestrian bridge, the results of the application of wavelet analysis were translated to reveal the characteristics of the oscillations of a non-separable steel three-span structure that received dynamic load impulses and determine the natural frequency of oscillations of the span structures. The results of the work are recommended for use during the examination and testing of bridges, the girder structures of which have hidden defects and damage, as well as for constant monitoring of the structures.

Natural and forsed osculations of pipelines in contact with the Wincler medium

In this paper, the pipeline is modeled as a curved rod in contact with the Winkler medium. Linear oscillations of a curved viscoelastic rod lying on the Winkler base are considered. The general formulation of the problem of free oscillations of a spatially curved viscoelastic rod with variable parameters is reduced to a boundary value problem for a system of ordinary integro-differential equations of the 12th order with variable coefficients relative to eigenstates; it can be solved by the method of successive approximations. The relations allowing to present the solution of the boundary value problem for the rod in an analytical form are formulated. It is established that the dimensionless complex frequencies of natural oscillations of a spatially curved rod, while maintaining the elongation of the rod constant, do not depend on it. The Poisson's ratio has little effect on the dimensionless real and imaginary parts of the natural frequencies.

A comprehensive approach to the analysis of the static and dynamic strength of the rotor of the smoke cleaner

The work describes the process of assessing the static and dynamic strength of the rotor of an axial smoke exhauster with twin blades which allow significantly increasing its efficiency. To build a calculation model for the strength analysis, the geometric model was simplified by removing bolted connections, slicing small surfaces, as well as modeling welds in the form of roundings and chamfers. Such simplifications made it possible to build a high-quality finite-element mesh with the sufficient accuracy for performing engineering calculations. Calculation models include a finite-element model of one sector of cyclic symmetry of the impeller of the smoke exhauster, which is used to more accurately determine its stress-strain state and local maximum stresses in the places of their concentration in the root sections of the blades and in the places where they are connected by jumpers due to the action of centrifugal forces, aerodynamic loads on the blade and thermal loads due to wheel heating, as well as a finite-element model of the entire rotor with a less detailed meshing of the impeller, which in this case plays the role of a mass-inertia element, and a more detailed meshing of the shaft to analyze its static strength and reactions in bearing assemblies from the action of gravity and determine natural frequencies and mode shapes of rotor vibrations and the possibility of resonances with multiples of the frequency of excitating loads. The analysis of the static strength of the impeller and the rotor shaft of the smoke exhauster was performed by comparing the von Mises equivalent stresses with the allowable stresses, and the analysis of the dynamic strength of the smoke exhauster rotor was performed by determining the level of detuning of the natural frequencies of vibrations from multiples of the frequency of the excitating load, which in this case was assumed to be the centrifugal forces acting on the unbalance of the rotating rotor. A detailed analysis of the results of calculations made it possible to establish that the elements of the smoke exhauster rotor meet the requirements of static strength under the action of all possible types of loads, and the natural frequencies of the rotor oscillations have a sufficient level of detuning from multiples of the frequency of the excitating load. In this way, a conclusion was made about the static and dynamic strength of the rotor of the smoke exhauster, which affect its resource and durability of operation in the nominal mode.

The Impact of Changes in the Conditions of Fastening of Steel Supporting Structures of Nuclear Power Plants Equipment and Piping on Their Seismic Resistance

The seismic resistance of nuclear power plants equipment and piping is determined, inter alia, by the seismic resistance of their steel supporting structures. The linear-spectral method, which involves using the results of the modal analysis of the structure under consideration, is widely used to assess the seismic resistance of these supporting structures. During the modal analysis, the structure’s dynamic characteristics are researched (in particular, the modes and values of natural oscillation frequencies). The dynamic characteristics of steel supporting structures affect the number of seismic loads that will be transmitted to them during an earthquake. The value of dynamic characteristics, among other issues, is influenced by the conditions of the steel supporting structures fastening. Therefore, it is relevant to research the impact of changes in the conditions of fastening of steel supporting structures of nuclear power plant equipment and piping on their seismic resistance. The paper gives the results of the research of dynamic characteristics, as well as the stress-strain state of steel supporting structures of nuclear power plant equipment and piping during changes in the conditions of their fastening

The dynamical problem on acting distributed load on the elastic layer

The wave field of an elastic half-layer is constructed, when a dynamic normal load distributed over a rectangular area acts on upper face at the initial moment of time. The lower face of the half-layer is rigidly fixed to the foundation, and the side border is in the conditions of a smooth contact. The method of decomposing the system of motion equations into a system of equations and an independently solvable equation is used, this approach was proposed by Popov~G.~Ya. Laplace and Fourier integral transformations are applied directly to the motion equations and boundary conditions, which reduces the problem to a vector one-dimensional boundary value problem, which is solved by the matrix differential calculus method. The output displacements are obtained using inverse integral transformations. The case of steady oscillations was considered and the amplitude of vertical displacement occurring in the layer was analyzed depending on the shape of the distributed load section, the material of the layer medium and the values of the natural frequency of the layer oscillations.

Влияние конструкции разделяемого узла стыка между сетчатыми композитными адаптером и корпусом космического аппарата на собственные частоты их колебаний

Grid composite structures are widely used in rocket and space technology as the payload adapters and the hull load-bearing structures (HLBS) of a spacecraft. An important parameter for these structures, along with the strength characteristics, is the minimum natural oscillation frequency. The article considers layout of the Express-1000 satellite platform consisting of a grid composite adapter and the spacecraft grid composite HLBS with a separable junction between them. The paper studies the influence of the number of pyro-locks and the thickness of the frames on the natural oscillation modes of the adapter and the HLBS assembly. Calculation was carried out by the finite element method, the number of fixing points of the hull to the adapter and the thickness of the end frames were varying, and the forms of natural oscillations and their corresponding frequencies were determined. Using the obtained dependences, it becomes possible to determine the rational number of fixing points and the thickness of the HLBS and the adapter.

Prediction of impact of large constructions in gulf on bathymetry of its remote parts as inverse problem of wave theory

The intentional bottom changes associated with construction of dams, breakwaters and other large structures in gulfs (or harbors) often lead to unintentional delayed bottom changes at remote locations. These unintentional changes can occur due to adjustment of frequencies of natural water oscillations in gulfs (or harbors) to the frequencies of natural water oscillations in the connected seas. Affecting conditions for wave propagation, a large structure changes the natural frequencies of the gulf because these frequencies are predetermined by the gulf size and bathymetry. The sea impact will restore the old natural frequencies of the gulf, but such a restoration requires change the bottom and shoreline at various locations along the gulf. Prediction of these remote changes can be formulated as an inverse problem for shallow water equations. The key to this formulation is the orthogonality of eigen modes of water level variation and water flux in the shallow water theory. The computed predictions of bathymetry changes are compared with observed trends of changes in the Gulf of Finland.

МЕХАНИЧЕСКИЙ СТАБИЛИЗАТОР ВРАЩЕНИЯ

The study objective is to find a mechanical analogue of cyclotron motion and to determine the structure of the corresponding device, which is appropriately called a stabilized rotator. The topic of speed stabilization is relevant. With cyclotron motion, the Lagrangian of an electron is twice as large as its kinetic energy. In terms of macromechanics, this corresponds to the equality of kinetic and potential energies. This condition is key to the possibility of generalizing cyclotron motion to mechanics. It follows from this that the composition of a stabilized rotator should include elements that are able to store both of these energy types. Such elements are the load and the spring. The natural rotation frequency of the stabilized rotator is strictly fixed (it does not depend on either the moment of inertia or the angular momentum) and remarkably coincides with the natural frequency of the pendulum with identical parameters. When the angular momentum changes, the radius and tangential velocity change (the rotation frequency does not change and is equal to its own). At zero torque moment in stationary mode, the rotation frequency of the stabilized rotator cannot be arbitrary and takes a single value. Just as when the pendulum is forced to swing, the frequency does not coincide with its own frequency, the rotation frequency of the stabilized rotator does not coincide with its own rotation frequency when loaded. A stabilized rotor can be used to control the natural oscillation frequency of a radial oscillator, although in this case it may have strong competition with mechatronic systems. On the contrary, as a rotation stabilizer, its competitive capabilities are undeniable and determined by the extremely simple design.

Natural oscillation frequencies of a Rayleigh sphere levitated in standing acoustic waves.

Acoustic levitation is an important method of container-free processing, which counteracts gravity through exerting the acoustic radiation force on levitated objects. The Gorkov potential function is used to simplify the calculation of the acoustic radiation force acting on a Rayleigh sphere whose radius is much smaller than the wave length. For the case of a plane standing wave levitation system, a systematic analysis of the sphere dynamics is provided in the axial direction, assuming a small perturbation around the stable equilibrium locations. A generalized extension to an arbitrary standing wave field is provided, which gives formal expressions of the axial and transverse natural oscillation frequencies for the sphere. Particular emphasis is put on the natural oscillation frequencies with and without taking gravity into consideration. The computational results for Gauss and Bessel standing waves are provided as two special cases, which show that the transverse natural oscillation frequency will be overestimated when neglecting gravity, especially for a sphere with a relatively large density. Corresponding experiments are conducted to verify the dependence of the transverse natural oscillation frequency on the sphere density. The results obtained in this work are expected to provide a theoretical guide for enhancing the levitation stability and inversing the physical parameters from the sphere dynamics.

FREQUENCY PARAMETERS OF NATURAL OSCILLATIONS IN PLATE STRUCTURES TO DETECT PRESET INADMISSIBLE CRACK SIZES

The method for determining the parameters of natural oscillations of an object under test is offered, which allows detecting crack sizes in structures made of plates. The use of this method is particularly relevant for monitoring large structures that takes much time if other NDT types are used. The relation between the natural oscillations parameters and crack sizes is built on the use of the mathematical model of the energy carrier of free oscillations, and on dynamic crack theory. Resultant diagrams are made for various deformation types on crack edges, to be used to determine wave length and frequency of natural oscillations in a plate against a preset inadmissible crack length, thus enabling crack detection. To ensure efficient non-destructive testing, an example of essential set of forms of natural oscillations for a plate structure is given.

Determining features of the stressed state of a passenger car frame with an energy-absorbing material in the girder beam

The object of this study is the processes related to the emergence, perception, and redistribution of loads in the improved structure of a passenger car frame. The scientific and applied task tackled in this paper is to ensure the strength of the supporting structure of a passenger car under operating loads. In this regard, it is proposed to improve the frame of a passenger car by constructing a girder beam from two rectangular pipes filled with material with energy-absorbing properties. The regularities of the frame load have been determined by taking into consideration the proposed solutions. It was found that the maximum equivalent stresses in the frame, taking its improvement into account, are 11.2 % lower than in the structure without filler, and 11.7 % lower than in the typical design. The results reported here are explained by the fact that the use of rectangular pipes filled with energy-absorbing material contributes to an increase in the moment of resistance of the frame, and, accordingly, reduces stresses. In addition, the study has determined the natural oscillation frequencies of the frame. The results of the calculation of the strength of the weld in the zone of interaction of the girder beam with the pivot beams are given. A feature of the results obtained is that the improvement in the strength of the frame is achieved not by strengthening its components but reducing the load. The scope of practical application of the reported results concerns railroad transportation, as well as other sectors of mechanical engineering. The conditions for the practical use of these findings are the introduction of closed profiles in the structure of vehicles at the stage of their design and modernization. This study could help reduce the cost of maintaining passenger cars and improve the efficiency of their operation. In addition, the research might prove useful for designing modern railroad car structures

Load of the wagon-platform for transportation of bulk cargoes

Purpose. To substantiate the improvement of the load-bearing element of the wagon-platform for the possibility of transporting bulk cargoes. Methodology. In order to be able to transport bulk cargo on the wagon-platform, it is proposed to install a composite boiler module on it. In order to determine the dynamic load of the improved load-bearing structure of the wagon-platform, mathematical modeling was performed. The mathematical model formed by professor Bohomaz H.I. was used. However, within the framework of the research this model was refined to determine the load of the wagon-platform of the proposed design. The solution of the system of differential equations is carried out in the MathCad software package. To do this, the mathematical model was reduced to the normal Cauchy form, and then integrated by the Runge-Kutta method. The obtained acceleration is taken into account when calculating the strength of the advanced load-bearing structure of the wagon-platform. The calculation is performed in the SolidWorks Simulation software package, which implements the finite element method. Also, within the research the modal analysis of a load-bearing structure of the wagon-platform is carried out. Findings. Based on the calculations, it is established that the acceleration acting on the load-bearing structure of the wagon-platform car is 0.38 g, i.e. it is within acceptable limits. The results of the calculation of the strength of the improved design of the wagon-platform showed that the maximum equivalent stresses occur in the area of interaction of the spine beam frame with the pivot and is about 340 MPa, the maximum displacement made 8.6 mm. That is, the obtained stresses do not exceed the yield strength of the structural material. The results of the modal analysis showed that the values of the natural frequencies of oscillations are within acceptable limits, because the first natural frequency has a value greater than 8 Hz. Originality. The scientific substantiation of improvement of a load-bearing structure of a universal wagon-platform to transportations of bulk cargoes is carried out. Practical value. The conducted research will promote increase in efficiency of operation of railway transport and creation of developments concerning planning of innovative designs of a rolling stock.

Peculiarities of fatigue fracture of high-alloyed heat-resistant steel after its operation in steam turbine rotor blades

The article focuses on the causes of failures of steam turbine rotor blades made of high-alloyed heat-resistant steel 15Kh11MF. Premature failure of blades is caused by technological defects (intensive corrosion-erosion wear of the blade edges, uneven structure and distribution of micro damages) that occurred because of imperfections of welding and/or surface hardening. Cracks initiated from these defects change the natural oscillation frequency of the blades, leading to their resonance and thus overload. The corrosive steam-water mixture in the phase transition zone contributes to the fracture process. Long-term operation of steel in steam turbines leads to the precipitation of carbides along the grain boundaries that weaken the cohesion between adjacent grains with subsequent pore formation and their merging into defects comparable to grain size. Fatigue crack growth resistance of steel reduces as a result of its operation. The effective threshold values of the SIF range decrease more significantly than the nominal ones due to the crack closure effect. It results from the increased roughness of the fracture surface caused by intergranular fragments protruding above the flat fatigue relief. They have been formed during steel’s operation along the damaged grain boundaries, and their number progressively increased with the operation time. To assess the current state of operated steel, the structural (the part of intergranular fragments S in the unit area of the fracture surface) and mechanical (a relative loss in fatigue crack growth resistance due to operation) indicators are proposed. The basic relationship between them makes it possible to substantiate the conditions for the safe operation of 15Kh11MF steel in STR blades.

Stabilized Rotator for Mechatronic Automatic Systems

The aim of the work is to find a mechanical analogue of cyclotron motion and to determine the scheme of the corresponding device, which is appropriate to call a stabilized rotator. From the key circumstance that determines the possibility of generalizing cyclotron motion to mechanics, which consists in the fact that the Lagrangian of an electron is twice as large as its kinetic energy, which, as applied to a stabilized rotator, should be interpreted as the equality of kinetic and potential energies, it follows that the composition of a stabilized rotator should include elements, which are able to store both of these types of energy, namely, the load and the spring. The natural frequency of rotation of a stabilized rotator is strictly fixed (it does not depend on either the moment of inertia or the moment of momentum) and remarkably coincides with the natural frequency of oscillations of a pendulum with identical parameters. When the angular momentum changes, the radius and tangential velocity change (the rotation frequency does not change and is equal to its own). The position of the load, in which its center of mass coincides with the axis of rotation, corresponds to a state of indefinite equilibrium. During rotation, the load can deviate with equal probability in any of the two directions and, accordingly, both compression and extension of the spring can develop. The state of indefinite equilibrium can be eliminated by providing the initial (static) displacement of the load and the initial deformation of the spring equal to it. Just as the frequency does not coincide with the natural frequency during forced oscillations of the pendulum, the rotation frequency of a stabilized rotator under loading does not coincide with the natural rotation frequency. At zero torque in the stationary mode, the rotational speed of the stabilized rotator cannot be arbitrary and takes on a single value. A stabilized rotator can be used to control the natural frequency of a radial oscillator, although in this capacity it may have strong competition from mechatronic systems. On the contrary, as a rotation stabilizer, its competitive capabilities are undeniable and are determined by the extreme simplicity of the design.

Research and determination of the acoustic parameters of the movement of a cavitation bubble in a liquid medium according to discrete and continuous models

In the work, the study and determination of the acoustic parameters of the movement of a cavitation bubble in a liquid medium according to discrete and continuous models was carried out. The research is based on the hypothesis that the determination of the effective parameters of the work process of acoustic processing is implemented by applying a transitional physical model from a discrete to a continuous type of processing of the technological environment. The obtained analytical dependences allow to calculate the amplitude of oscillations and the frequency of natural oscillations. With the help of the specified formulas, it is possible to determine the zones of amplification or attenuation of the amplitude of oscillations for different frequencies of oscillations. The proposed formula for determining the frequency of natural oscillations, which takes into account changes in the properties of the medium from homogeneous at the initial stage to the appearance of cavitation bubbles at the specified frequency of natural oscillations. Numerical values of intensity, pressure, amplitude of oscillations, velocity, acceleration, viscosity and maximum bubble radius are given. The obtained numerical values can be used in practical calculations of acoustic processing parameters of different nature and properties of technological environments.

Kinematic analysis and estimation of the frequency of natural oscillations of a planar lattice

Introduction. A statically determined regular symmetrical rectangular lattice is fixed on two supports. The mass of the lattice is distributed over its nodes. A kinematic analysis is presented that reveals the variability of the lattice for a certain number of panels. An algorithm for searching for the distribution of virtual velocities of lattice nodes in this case is described. A formula is derived for the dependence of the lower estimate of the first frequency of natural oscillations of the structure on the number of panels from the range of permissible values.
 
 Materials and methods. Calculation of forces in the bars of a statically determinate lattice is performed by the method of cutting out nodes using Maple computer mathematics system operators. To determine the rigidity of the structure, the Maxwel – Mohr formula is used on the assumption that the elastic moduli and cross-sectional areas of all lattice rods are the same.
 
 Results. It is shown that when the number of panels is a multiple of three, the determinant of the matrix of the system of equilibrium equations degenerates, and the system becomes an instantly variable mechanism. A corresponding picture of the distribution of knot velocities is given. Using the approximate Dunkerley estimate, a formula is derived for the dependence of the lower limit of the first oscillation frequency of the truss on the number of panels. The generalization of a series of particular solutions to an arbitrary number of panels is performed in the Maple system by induction.
 
 Conclusions. The proposed lattice model admits an analytical solution for the lower estimate of the first natural frequency. Comparison of the result with the numerical solution obtained for the lowest frequency of the entire frequency spectrum of natural lattice vibrations shows its high accuracy. It is shown that the error of the found analytical estimate decreases with an increase in the number of panels. The solution can be used in optimization problems and for a preliminary assessment of the designed structure.

Speed Self-Stabilization

The aim of the work is to find a mechanical analogue of cyclotron motion and to determine the scheme of the corresponding device, which is appropriate to call a stabilized rotator. From the key circumstance that determines the possibility of generalizing cyclotron motion to mechanics, which consists in the fact that the Lagrangian of an electron is twice as large as its kinetic energy, which, as applied to a stabilized rotator, should be interpreted as the equality of kinetic and potential energies, it follows that the composition of a stabilized rotator should include elements, which are able to store both of these types of energy, namely, the load and the spring. The natural frequency of rotation of a stabilized rotator is strictly fixed (it does not depend on either the moment of inertia or the moment of momentum) and remarkably coincides with the natural frequency of oscillations of a pendulum with identical parameters. When the angular momentum changes, the radius and tangential velocity change (the rotation frequency does not change and is equal to its own). The position of the load, in which its center of mass coincides with the axis of rotation, corresponds to a state of indefinite equilibrium. During rotation, the load can deviate with equal probability in any of the two directions and, accordingly, both compression and extension of the spring can develop. The state of indefinite equilibrium can be eliminated by providing the initial (static) displacement of the load and the initial deformation of the spring equal to it. Just as the frequency does not coincide with the natural frequency during forced oscillations of the pendulum, the rotation frequency of a stabilized rotator under loading does not coincide with the natural rotation frequency. At zero torque in the stationary mode, the rotational speed of the stabilized rotator cannot be arbitrary and takes on a single value. A stabilized rotator can be used to control the natural frequency of a radial oscillator, although in this capacity it may have strong competition from mechatronic systems. On the contrary, as a rotation stabilizer, its competitive capabilities are undeniable and are determined by the extreme simplicity of the design.

The Power Transformer Winding Input Admittance and Frequency Response

The frequency response analysis is increasingly more widely used to estimate the electrical and mechanical condition of power transformer windings. The frequency response measurement procedure has been prescribed in the existing international and national standards. However, the interpretation of measurement results still remains a subject of discussion. Various approaches have been used to interpret the results, including an analysis of transformer windings’ complex impedances and admittances obtained by calculations from frequency responses. To improve the frequency response interpretation, the article presents an analysis of power transformer winding input and output currents and admittances based on simplified equivalent circuits. Qualitative and quantitative results are presented, which show that the admittances obtained from measured frequency responses differ essentially from the true input admittances of the windings. The main factors causing the differences between these winding admittances are considered. It is shown that despite the existing differences, the current through the neutral and the winding admittance determined from the measured frequency response can also be used in practice to determine the power transformer winding natural oscillation frequencies.

Research of the Stress-Strain State for Steel Support Structures of Nuclear Power Plant Components under Seismic Loads

Seismic resistance of equipment and piping of nuclear power plant units is determined, particularly, by seismic resistance of their steel support structures. A significant number of these structures are located in the reactor building of nuclear power plants and belong to seismic resistance categories I and II. Therefore, such support structures of nuclear power plant equipment and piping, in general, shall perform their functions under seismic hazards that correspond to the safe shutdown earthquake and design-basis earthquakes, respectively.
 Steel support structures of serial equipment (piping valves, electric and pneumatic drives of piping and pump valves, expansion tanks, etc.) are often designed “on site” in the limited conditions of their actual location. Therefore, it is necessary to design steel support structures of a number of equipment taking into account the fact that it can be connected, in particular, to existing piping with their actual routing. In connection with the above, the research of operation for such support structures under seismic loads is relevant.
 The article presents the research results for the spectrum of natural oscillation frequencies of the equipment support structure, and the stress-strain state of the considered structure under seismic loads is determined. At the same time, the parameters during normal operation and maximum design basis accident at the nuclear power plant power unit are considered. The calculated combination of loads, which simultaneously includes two episodic impacts (maximum design basis accident and an earthquake) is considered.