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.

Random oscillations of nonlinear systems with distributed Parameter

The article analyzes random vibrations of nonlinear mechanical systems with distributed parameters. The motion of such systems is described by nonlinear partial differential equations with corresponding initial and boundary conditions. In our case, the system as a whole is limited, so any motion can be considered as the sum of the natural oscillations of the system, i.e. in the form of an expansion of the boundary value problem in terms of own functions. The use of the theory of random processes in the calculation of mechanical systems is a prerequisite for the creation of sound design methods and the creation of effective vibration protection devices, these methods allow us to investigate dynamic processes, to determine the probabilistic characteristics of displacements of points of the system and their first two derivatives. In the work established these conditions are met, they provide effective vibration protection of the system under study with wide changes in the pass band of the frequencies of the random vibration effect, and the frequency of the disturbing force is much greater than the natural frequency of the system as a whole, in addition, with an increase in the damping capacity of the elastic-damping link of the system, the intensity of the random process significantly decreases, which in turn leads to a sharp decrease in the dynamic coefficient of the system.

On the assessment of the potential for extracting information on defects in the main pipeline by the method of radar analysis of natural oscillation frequency

The paper deals with the issue of diagnostics of main pipelines by the vibration method, exploring the influence of the formation of cracks in the walls of the pipeline on the change in the frequency characteristics of the pipeline. By analyzing the difference in the natural oscillation frequency of a pipe with a crack and without it, it is possible to assess the location of the defect with a certain accuracy. The issue of the possibility of reading the natural oscillations of the main pipeline from the surface of the soil located above the pipeline is also being studied.

Testing the CMIP6 GCM Simulations versus Surface Temperature Records from 1980–1990 to 2011–2021: High ECS Is Not Supported

The last-generation CMIP6 global circulation models (GCMs) are currently used to interpret past and future climatic changes and to guide policymakers, but they are very different from each other; for example, their equilibrium climate sensitivity (ECS) varies from 1.83 to 5.67 °C (IPCC AR6, 2021). Even assuming that some of them are sufficiently reliable for scenario forecasts, such a large ECS uncertainty requires a pre-selection of the most reliable models. Herein the performance of 38 CMIP6 models are tested in reproducing the surface temperature changes observed from 1980–1990 to 2011–2021 in three temperature records: ERA5-T2m, ERA5-850mb, and UAH MSU v6.0 Tlt. Alternative temperature records are briefly discussed but found to be not appropriate for the present analysis because they miss data over large regions. Significant issues emerge: (1) most GCMs overestimate the warming observed during the last 40 years; (2) there is great variability among the models in reconstructing the climatic changes observed in the Arctic; (3) the ocean temperature is usually overestimated more than the land one; (4) in the latitude bands 40° N–70° N and 50° S–70° S (which lay at the intersection between the Ferrel and the polar atmospheric cells) the CMIP6 GCMs overestimate the warming; (5) similar discrepancies are present in the east-equatorial pacific region (which regulates the ENSO) and in other regions where cooling trends are observed. Finally, the percentage of the world surface where the (positive or negative) model-data discrepancy exceeds 0.2, 0.5 and 1.0 °C is evaluated. The results indicate that the models with low ECS values (for example, 3 °C or less) perform significantly better than those with larger ECS. Therefore, the low ECS models should be preferred for climate change scenario forecasts while the other models should be dismissed and not used by policymakers. In any case, significant model-data discrepancies are still observed over extended world regions for all models: on average, the GCM predictions disagree from the data by more than 0.2 °C (on a total mean warming of about 0.5 °C from 1980–1990 to 2011–2021) over more than 50% of the global surface. This result suggests that climate change and its natural variability remain poorly modeled by the CMIP6 GCMs. Finally, the ECS uncertainty problem is discussed, and it is argued (also using semi-empirical climate models that implement natural oscillations not predicted by the GCMs) that the real ECS could be between 1 and 2 °C, which implies moderate warming for the next decades.

Enhanced Feature Extraction Algorithms Using Oscillatory-Mode Pulsed Eddy Current Techniques for Aircraft Structure Inspection

A review of the existing literature shows that modern pulsed eddy current (PEC) technique for flaw detection in aircraft structure inspection is typically carried out in aperiodic mode. Аt the same time, the unstable characteristic points of the EC signal usually used as informative parameters can restrict the potential of this excitation mode due to significant measurement errors. This article considers an advanced PEC method of NDT based on the oscillatory mode. To obtain the conditions concerned with different modes of EC probe response oscillations, an equivalent scheme of the “testing object – EC probe” system was developed and analyzed. The frequency and attenuation coefficient of natural oscillations are proposed as the informative parameters of the probe signals. The obtained mathematical model of the probe signals allows for the dependence of proposed signal parameters on the characteristics of the testing object to be evaluated. Herein, we first develop algorithmic software for determining and analyzing the discrete amplitude and phase characteristics of PEC NDT signals based on the simulation results. The errors of the natural frequency oscillations and the attenuation coefficient determination as well as the optimal time for its determination are analyzed in order to minimize the possible errors. Next, the proposed informative parameters are experimentally investigated using a set of specimens. The obtained results confirm the possibility of the proposed methodology to enhance the inspection procedures related to the electrical conductivity and geometric parameters measurements as well as the detected defect sizing.

Ship control in storm conditions

Приведены выражения для определения периодов собственных поперечных и продольных колебаний судна, как точные, так и приближённые, но в тоже время достаточные для их практического использования на судне. Представлены формулы для расчёта поперечной метацентрической высоты после принятия груза судном перед выходом в море. Выведены формулы для определения критических зон резонансной качки по крену и дифференту, как по скорости судна, так и по курсовому углу по отношению к направлению распространения штормового волнения моря. Представлены формулы для определения кажущегося периода встречи судна с волной, которые являются основой для расчёта резонансных зон. Выведенные соотношения для определения зоны резонанса по скорости судна при заданном курсовом угле и по курсовому углу при заданной скорости судна представлены при условии известного периода штормового волнения моря и курсового угла судна по отношению к направлению распространения волнения моря. Приведены формулы для определения амплитуды качки в условиях резонанса, если отношение периода собственных колебаний судна к кажущемуся периоду волны находится в пределах 0,7 – 1,3. Представлены выражения для определения амплитуд качки по крену и дифференту, вызывающие морскую болезнь у персонала, а также критические значения боковых перегрузок, влияющих на правильность его действия. Expressions for determining the periods of the natural transverse and longitudinal vibrations of the vessel, both exact and approximate, are given, but at the same time sufficient for their practical use on the vessel. The formulas for calculating the transverse metacentric height after the cargo has been accepted by the vessel before going to sea are presented. Formulas are derived for determining the critical zones of resonant pitching in terms of roll and trim, both in terms of the ship's speed and in the heading angle in relation to the direction of propagation of storm waves of the sea. The formulas for determining the apparent period of the ship's meeting with the wave are presented, which are the basis for calculating the resonance zones. The derived relations for determining the resonance zone by the speed of the vessel at a given heading angle and by the heading angle at a given speed of the vessel are presented under the condition of a known period of stormy sea waves and the heading angle of the vessel in relation to the direction of propagation of sea waves. Formulas are given for determining the amplitude of pitching under resonance conditions if the ratio of the period of natural oscillations of the vessel to the apparent period of the wave is within 0.7 - 1.3. Expressions for determining the amplitudes of roll and pitch that cause motion sickness in personnel, as well as the critical values of lateral g-forces that affect the correctness of its action, are presented.

Дифференциальный фон электрического сигнала, снимаемого с индукционной магнитной головки

Experimental dependences U(t) of the electric voltage taken from an induction magnetic head (MH) moving relative to a magnetic carrier (MH) are presented. The backgrounds of the edges of the MN, local defects of the MN, the background of the inhomogeneous magnetic field of the local source, the backgrounds of defects and structural inhomogeneities of the object, the etheric electromagnetic background, the background of the quality of the surface of the object and surface microscopic inhomogeneities of the material have been identified and investigated. The resonant backgrounds of self-excitation of the measuring system on the signals of the edges of the MN, defects of the MN, instrument and network pickups and interference, object defects, and etheric electromagnetic fields are revealed and investigated. Resonance peaks are the result of self-excitation of the measuring system, which includes the MG, and arise on the trailing edges of any signals of sufficient magnitude, the duration of the trailing edge of which is about a quarter of the period of natural oscillations of the measuring system. The amplitude and frequency spectra of the background signals of object defects, MI and noise and the analytical expressions describing them are determined. The results of the extraction of the useful signal from the complete signal recorded on the MN are shown. Investigations of the differential background of an electric signal allow, together with the previously developed methods of hysteresis interference, to control the properties of objects in an automatic mode with program control, which significantly increases the sensitivity and accuracy of control. To achieve this goal, it is recommended to set the parameters of the measurement system at the threshold of the onset of natural free oscillations in it.

Application of mathematical modeling methods to determine the effect of soil on natural vibration frequencies of pipelines

When a pipeline is subjected to an external influence that can affect the frequency of its natural vibrations, the parameters of its natural vibrations change, which increases the measurement error, and often simply distorts the results of vibration control. For pipelines, such an impact may be the influence of the soil when pipelines are laid without a channel. Different types of soil affect the change in the natural vibration frequency of the pipeline in different ways.The purpose of the article is to analyze the influence of various types of soils on the parameters of natural vibrations of the pipeline. The aim of the study was to theoretically confirm the dependence of the change in the frequency of vibration of the pipeline under the influence of soil.A modal analysis of natural oscillations of 5 polyethylene pipelines was performed. As initial data, it is assumed that the design pipeline is laid in a trench with inclined walls, with the slope laid on a flat base at a depth of 2.5 m. The calculations were performed in the ANSYS finite element analysis software package. In order to build a mathematical model, the degree of soil impact on pipelines is determined by studying the vertical and lateral pressure of the soil on the above pipelines, and the natural vibrations of pipelines are analyzed.The results of the modal analysis for polyethylene pipes with a laying scheme with inclined walls and different soils (gravel sands, coarse and medium-sized; heavy clays) are presented. The soils were chosen that are the most common on the territory of Russia.Thus, the obtained dependence of the degree of influence of different soil on the natural frequencies of pipelines significantly increases the reliability of vibration diagnostics of buried communications, can facilitate the work on its organization and allow determining long-term forecasts of pipeline operation.

SPECTRUM OF OWN FREQUENCIES OF A SPATIAL SURFACING GIRDER

Statement of the problem. The scheme of a statically definable girder of a spatial rectangular surfacing is discussed. The problem is to identify the formula for the dependence of the lower estimate of the first frequency of the natural oscillations of the structure by means of the Donkerley method on the number of panels. The truss has supports on the sides and consists of separate rod cells connected in pyramids. Results. Based on the analysis of the sequence of analytical solutions for the first frequency of girders with a different number of panels by induction, the coefficients in the desired formula are derived. The common members of the sequences of coefficients are found as solutions of homogeneous recurrent equations formed according to the results of the calculations using Maple operators. The resulting dependences are obtained in the form of polynomials by the number of panels. A comparison of the analytical solution with the numerical one is provided.Conclusions. An algorithm for deriving an analytical estimate of the fundamental frequency of oscillations of a spatial structure depending on the number of panels, mass, size, and elastic properties of the material is shown. The spectrum of oscillation frequencies of the structure is analyzed. The resulting dependences can be employed in seismic and structural optimization problems.

Analytical study of the natural bending oscillations of a concave beam with parabolic change in thickness

The synthesis of factorization and symmetry methods produced a general analytical solution to the fourth-order differential equation with variable coefficients. The form and structure of the variable coefficients correspond, in this case, to the problem of the oscillations of a concave beam of variable thickness. The solution to this equation makes it possible to study in detail the oscillations of such and similar, for example convex, beams at the different fixation of their ends' sections. A practical confirmation has been obtained that the beam whose thickness changes in line with the concave parabola law H=a2x2+1, where a is the concave factor, demonstrates an increase in the natural frequencies of its free oscillations with an increase in its rigidity. As an example, the object's maximum deflection dependence on the beam rigidity factor has been established. The nature of this dependence confirmed the obvious conclusion that the deflections had decreased while the rigidity had increased. The evidence from the calculation results can be a testament to the correctness of the reported procedure of problem-solving. The considered problem and the analytical solution to it could serve as a practical guide to the optimal design of beam structures. In this case, it is very important to take into consideration the place and nature of the distribution of cyclical extreme operating stresses. The resulting ratios to solve the problem make it possible to simulate the required normal stresses in both the fixation and central zones when the rigidity parameter is changed. Designers could predict such a parabolic profile of the beam, which would ensure the required reduction of maximum stresses in the place of fixing the beam. The considered example of solving the problem of the natural oscillations of the beam with rigid fixation of the ends illustrates the effectiveness of the factoring and symmetry methods used. The developed solution algorithm could be extended to study the natural bending oscillations of the beam at other fixing techniques, not excluding a variant of a completely free beam