Frequency Of Free Oscillations Research Articles

Muzzle bubble dynamics characterization of underwater launching

To comprehensively understand the dynamic behavior of muzzle bubbles during underwater launching, an emptying process aligned with the muzzle flow characteristics is established and an evaporative condensation mechanism is modeled according to the high temperature and pressure properties of the propellant gas. Utilizing the spherical bubble theory, which comprises the inflation process and evaporative condensation effects, the dynamics of muzzle bubbles and their corresponding pressure waves are investigated. The numerical simulation results well agree with the experimental observations in terms of bubble radius and near-field pressure waves. Furthermore, the influence of two key factors on the bubble dynamics is examined: underwater launching depth and initial muzzle pressures. The results illustrate that the inflation process needs to be accurately described for precise pressure wave predictions. Using the evaporation condensation model, the bubble radius and frequency can be accurately characterized. Moreover, the launching depth influences the free expansion radius and oscillation frequency mostly due to the increase in hydrostatic pressure, which decreases by 33% and increases by 150% in the 1–20 m range, respectively. The initial muzzle pressure affects the initial expansion velocity and initial shock wave mainly due to the increase in the mass flow rate, which increase by 56% and 82% in the 35–65 MPa range, respectively.

Application of Maple Mathematical Package to Search for Transfer Gear Oscillation Frequencies

Considers direct spectral problem of determination of frequencies of free torsional oscillations of transfer gearbox, component part of power plants with diesel fuel, gearboxes. The gearbox is modeled by a dynamic mechanical system with three degrees of freedom of bearings and gear. The frequency equation of the direct problem was found, according to which numerical calculations were carried out to search for the frequencies of torsional vibrations of the transfer gearbox. To the conclusion of the equation and the search for the frequencies of the gearbox oscillations, a software implementation of transformations using the functionality and libraries of the Maple mathematical package is given.

Formulas for Calculating Deformations and Natural Frequency of Free Vibrations of a Hexagonal Tower

Statement of the problem. A scheme of a six-sided prismatic statically determinate spatial girder is proposed. The task is set by induction to derive formulas for the dependence of the deflection of the structure and the lower limit of the main frequency of natural vibrations on the number of panels along the height of the prism. Materials and methods. The forces in the rods along with the reactions of the supports are found in an analytical form by means of the method of cutting nodes in the Maple symbolic mathematics system. One of the nodes at the base of the truss has a spherical support, one has a cylindrical support, the remaining four supports are racks. The top deflection is determined by the Maxwell-Mohr formula. From the analysis of the sequences of coefficients in the formulas for individual structures with a different number of panels, their common members are determined, which are included in the desired calculation formula. The Dunkerley method is used to estimate the first frequency of free oscillations. Results. For various types of loads, formulas for the dependence of girderf deflections on the number of panels are obtained. The coefficients in the solution are polynomial in the number of panels. The derived analytical dependence of the first frequency on the number of panels in comparison with the numerical solution has a small error, which decreases with increasing number of panels. Conclusions. A design of an axisymmetric statically determinate tower-type girder has been developed, which allows analytical solutions to the problem of deflection and the problem of the first natural frequency for an arbitrary number of panels. The resulting formulas can be used to assess the accuracy of numerical solutions and for preliminary calculations of models of structures of this type.

Variation-Difference Method of Calculation of Layered Rubber-Metal Vibration Isolators Used for Protection of Reinforced Concrete Buildings from Anthropogenic Vibration

In the modern construction complex of the city of Moscow for the protection of buildings and structures from man-made vibration arising from the movement of trains of rail transportation (subway trains, railroad lines and trams). To protect buildings and structures from anthropogenic vibration arising from the movement of rail transport trains (subway trains, railroad lines and streetcars), Moscow uses layered rubber-metal vibration isolators [1]. Most often, to determine their static and dynamic characteristics, the finite element method (FEM) is used, which makes it possible to determine all components of the stress-strain state and frequencies of free oscillations in the loaded state practically for any structural forms of isolators. However, for the most popular software packages that implement FEM, the problem of optimizing the structural shape of the vibration isolator still requires significant time expenditures for multiple changes of the finite element mesh, repeated setting of boundary conditions and implementation of a series of calculations. Only some of the software packages implementing the FEM solve optimization problems of the shape of the product being calculated, most often it is related to foreign software products with universal functionality. Variation-difference method (VDM) is the closest to the finite element method (FEM) in terms of its computational capabilities. It is possible to create program modules that repeatedly and automatically solve three-dimensional problems of elasticity theory taking into account the changed geometry of the vibration isolator: the dimensions of the product, the location of perforations within the rubber layers, as well as the thickness of the rubber layer and other parameters important for obtaining an effective technical solution for vibration isolation of buildings. Further, the article describes the method of implementation of the variational-difference method (VDM) as applied to the solution of the problem of determining the components of the stress-strain state inside a three-dimensional layered vibration isolator with perforations of different sizes having different locations relative to the contour of the vibration isolator, i.e., the solution of the problem of optimizing the three-dimensional shape of the vibration isolator is given.

NONLINEAR EFFECTS IN VISCOELASTIC DROP SHAPE OSCILLATIONS

A study of axisymmetric shape oscillations of viscoelastic drops in a vacuum is conducted, using the method of weakly nonlinear analysis. The motivation is the relevance of the shape oscillations for transport processes across the drop surface, as well as fundamental interest. The study is performed for, but not limited to, the two-lobed mode of initial drop deformation. The Oldroyd-B model is used for characterizing the liquid rheological behavior. The method applied yields a set of governing equations, as well as boundary and initial conditions, for different orders of approximation. In the present paper, the equations and solutions up to second order are presented, together with the characteristic equation for the viscoelastic drop. The characteristic equation has an infinite number of roots, which determine the time dependency of the oscillations. Solutions of the characteristic equation are validated against experiments on acoustically levitated individual viscoelastic aqueous polymer solution drops. Experimental data consist of decay rate and oscillation frequency of free damped drop shape oscillations. With these data, solutions of the characteristic equation dominating the oscillations are identified. The theoretical analysis reveals nonlinear effects, such as the excess time in the prolate shape and frequency change for varying initial deformation amplitude. The influences of elasticity, measured by the stress relaxation and deformation retardation time scales, are quantified, and the effects are compared to the Newtonian case in the moderate-amplitude regime.

Probing signals of atmospheric gravity waves excited by the July 29, 2021 MW8.2 Alaska earthquake

It is commonly believed that the atmosphere is decoupled from the solid Earth. Thus, it is difficult for the seismic wave energy inside the Earth to propagate into the atmosphere, and atmospheric pressure wave signals excited by earthquakes are unlikely to exist in atmospheric observations. An increasing number of studies have shown that earthquakes, volcanoes, and tsunamis can perturb the Earth's atmosphere due to various coupling effects. However, the observations mainly focus on acoustic waves with periods of less than 10 min and inertial gravity waves with periods of greater than 1 h. There are almost no clear observations of gravity waves that coincide with observations of low-frequency signals of the Earth's free oscillation frequency band within 1 h. This paper investigates atmospheric gravity wave signals within 1 h of surface-atmosphere observations using the periodogram method based on seismometer and microbarometer observations from the global seismic network before and after the July 29, 2021 MW8.2 Alaska earthquake in the United States. The numerical results show that the atmospheric gravity wave signals with frequencies similar to those of the Earth's free oscillations 0S2 and 0T2 can be detected in the microbarometer observations. The results confirm the existence of atmospheric gravity waves, indicating that the atmosphere and the solid Earth are not decoupled within this frequency band and that seismic wave energy excited by earthquakes can propagate from the interior of the Earth to the atmosphere and enhance the atmospheric gravity wave signals within 1 h.

ЗАВИСИМОСТЬ ОСНОВНОЙ ЧАСТОТЫ СОБСТВЕННЫХ КОЛЕБАНИЙ ПЛОСКОЙ ШПРЕНГЕЛЬНОЙ ФЕРМЫ ОТ ЧИСЛА ПАНЕЛЕЙ

A scheme of a statically determined planar truss with downward braces and truss reinforcement is proposed. The problems of the lower and upper boundaries of the main natural frequency of free oscillations of the structure are solved. A model with masses concentrated at the nodes is adopted. To find the formula for the dependence of the oscillation frequency limits on the number of panels, the Dunkerley and Rayleigh methods are used. The expressions for the forces in the truss rods are found from the solution of the system of linear equilibrium equations for all nodes. To obtain the matrix of the system of equations, Maple computer mathematics operators are used. The rigidity of the structure is calculated using the Maxwell – Mohr formula. The sequence of solutions obtained for trusses with different numbers of panels is generalized by induction to an arbitrary case. For the coefficients of the formula in the desired solution, homogeneous linear recurrent equations are compiled and solved. The obtained solution is compared with the numerical solution of the problem of the oscillation spectrum of a structure with many degrees of freedom. It is shown that the first frequency of the spectrum is close to its analytical estimate. With an increase in the number of panels in the truss, the accuracy of the derived formulas increases.

FREE VIBRATIONS OF THREE BOUND WEIGHTS WITHOUT THE USE OF POTENTIAL ENERGY

The possibility of occurrence of free sinusoidal linear oscillations in a mechanical monoreactive oscillator with three weights is considered. The main difference between such an oscillator and a classical pendulum is the exclusion of potential energy from energy exchange, since the considered oscillator does not contain elastic elements. As a result, free sinusoidal oscillations are caused by the mutual transformation of the kinetic energy of one of the loads into the kinetic energy of the other two loads. The use of not two, but three weights ensures the symmetry of the oscillator circuit and, as a result, the balance of the mechanism. The absence of elastic elements excludes the elasticity coefficient from the mathematical description, as a result of which the natural frequency of free sinusoidal oscillations of a mechanical monoreactive oscillator with three weights is not limited to a fixed value and is determined solely by the initial conditions.

Improvement and research of a suspension-shock absorber for attaching a hydraulic hammer to an excavator

Problem. The article proposes a developed construction of a suspension – shock absorber, which made it possible to install a hydraulic hammer as a working equipment, simultaneously isolating it from vibrations.
 A mathematical model of the system was developed, the basis of which is a suspension – shock absorber for attaching a hydraulic hammer to an excavator, which includes modeling of its mechanical system. Goal. The goal is the development and research of a sub-axle - shock absorber for the attachment of hydraulic hammer equipment (hydraulic hammer) to an excavator. Methodology. The solution of the set problems involved the creation of a mathematical model of the vibration system and its implementation using the MathCAD application program. Originality. It is proposed to transform the differential equations of the mathematical model of the vibration system by the method of decreasing order and as a result obtain the displacements, velocities, and accelerations of individual elements and obtain the nature of the change in the disturbing force of the vibrating hammer. Using the MathCAD program, a study of the dynamics of the system, which includes a hydraulic ham a shock absorber and an excavator, was performed, as well as the frequencies of free oscillations of the system in question were determined. Practical value. The obtained results of scientific research on vibration protection of excavators, which are operated with attached shock devices (hydraulic hammers), with the use of the MathCAD mathematical program, can be successfully used in the development, as well as in the determination of dynamic loads of similar vibration protection systems used in industry.

ФОРМУЛА ДЛЯ НИЖНЕЙ ОЦЕНКИ ПЕРВОЙ ЧАСТОТЫ СОБСТВЕННЫХ КОЛЕБАНИЙ ФЕРМЫ ТРЕХГРАННОЙ БАШНИ

A scheme of a statically determinate tower-type spatial truss is proposed. Three lateral faces of the structure have double diagonal lattices, from below the truss rests on six racks and three additional horizontal lateral braces. The problem of the first natural frequency of free oscillations of the structure is solved. The inertial properties of the truss are modeled by the masses in its nodes. Each mass has two horizontal degrees of freedom. To obtain an analytical expression for the dependence of the lower limit of the oscillation frequency on the number of panels, the approximate Dunkerley method is used. The forces in the truss rods are found from the solution of the system of equilibrium equations for the nodes. The matrix of the system of equations is compiled in the Maple computer mathematics system. The rigidity of the structure is determined using the Mohr integral. A series of solutions obtained for a different number of panels is generalized to an arbitrary case by induction. For sequences of coefficients in the solution, linear recurrent equations are compiled and solved. The analytical solution is compared with the numerical solution of the problem of the vibration spectrum of the structure. It is shown that the first frequency of the spectrum is higher than the analytical estimate by no more than 30%. The accuracy of the estimate depends to a greater extent on the number of panels. As the number of panels increases, the accuracy of the resulting estimate increases. A trusses with only an even number of panels are considered. It is noticed that with an odd number of panels, the determinant of the system of equilibrium equations of nodes vanishes, which indicates the kinematic degeneration of the structure.

Mixed forms of free oscillations of a rectangular CFCF-plate

Mixed (symmetrically/antisymmetric) forms of natural oscillations of a thin rectangular plate of constant thickness, in which two parallel sides are rigidly pinched, and the other two are free (CFCF plate, C — clamped, F — free), are studied. When all the conditions of the boundary value problem are satisfied, a resolving infinite homogeneous system of linear algebraic equations with respect to unknown coefficients of the series is obtained using two hyperbolo-trigonometric series of the coordinate deflection function. Even functions on one coordinate and odd functions on another coordinate were used to obtain symmetric-antisymmetric waveforms. As a parameter, the resulting system contains the relative frequency of free oscillations. Nontrivial solutions of the reduced system were found by the method of successive approximations in combination with a search of the frequency parameter. Numerical results are obtained for the spectrum of the first six mixed (symmetric/antisymmetric — S-A and A-S) forms of free oscillations of a thin square CFCF plate of constant thickness. The natural frequencies were compared with the results of other authors and with known experimental values. The influence on the accuracy of the results of the number of members held in rows (the size of the reduced system) and the number of iterations is investigated. 3-D images of the found waveforms are presented. The results obtained can be used in the design of various sensors and sensors using the resonance phenomenon.

Experimental determination of dynamic characteristics of a railway track

It is known a priori that vibrations that occur in the elements of the track structure during the passage of a train load are a superposition of free and forced mechanical vibrations. It has been established that the range of oscillation frequencies of the elements of the track superstructure includes oscillations with a frequency from tens to hundreds and thousands of hertz. However, the influence of vibrations on the track and their dependence, in turn, on the design of the track, has not been fully studied, which causes controversy between specialists in this matter. There is an opinion that in intermediate fastenings, the main role is played by an elastic gasket, which ensures the vertical rigidity of the rail-sleeper assembly. It has been experimentally established that in the frequency range below the frequency of free oscillations of rail fastenings, the force applied to the upper surface of the elastic gasket is transferred to the lower surface in an unchanged form.
 The article discusses the vibrations that occur in the elements of the upper structure of the track (rails, sleepers, rail fastenings, ballast base) during the passage of a moving load, which significantly affect the strength, and, consequently, the durability of both the elements themselves and the railway track generally. Vibrograms, oscillograms, accelerograms of rail oscillations and their spectra during the passage of TALGO are presented. At present, the VOSSLOH fastening, developed in Germany, has been widely used in the construction of tracks on high-speed railways of the Republic of Kazakhstan. In this regard, there is a need for a comprehensive study of the operation of this type of fastening under a train load

Development of a mathematical model of a frequency-controlled electromagnetic vibration motor taking into account the nonlinear dependences of the characteristics of the elements

The wider application of vibration machines (VM) with electromagnetic motors (EMVM) in various industries, including in vibration test benches for telecommunication devices and equipment has been studied in the paper. Difficulties associated with maintaining their productivity and efficiency at a given level are largely hindered. These factors mainly depend on the determination of the degree of influence of the nonlinearities of the input-output characteristics on the output values of the VM, the possibilities of tuning into the resonance mode, and the control of the output values while ensuring the energy-saving mode of operation of the EMVM. The mathematical model has been developed for a frequency-controlled EMVM taking into account the nonlinear dependencies of the characteristics of the elements, which makes it possible to determine the most accurate amplitude of oscillations of the EMVM working body when passing through resonance, observed with changes in the voltage frequency. The analysis of the physical processes of the influence of nonlinear elements of the electric circuit of the EMVM on the electromagnetic quantities and the derived analytical expressions of these nonlinearities, which make it possible to determine the harmonic composition of the current of the EMVM winding, are made. The system of vector control of the EMVM and the inclusion of a mathematical model of the engine into the control system have been developed, which will allow calculating the frequencies of free oscillations of the working body for tuning into the resonance mode.

Method of Preservation of Free Oscillation Frequencies of Elastically Interconnected Shafts

Method of Preservation of Free Oscillation Frequencies of Elastically Interconnected Shafts

DESIGNING OF OPTIMAL MOTION CONTROLS FOR FINITE RIGIDITY OBJECTS WITH MINIMUM ENERGY INTENSITY

An algorithm for equivalent replacement of a reversionally constructed optimal control with another control is used, provided that the energies are equal to achieve the goal of motion. It is shown that with an increase in the degree of the polynomial approximating the translational acceleration, not only does the energy decrease for the practical implementation of an equivalent control of the «acceleration-deceleration» type, but also (at a given frequency of free oscillations of a moving elastic object) a decrease in the time to reach absolute quiescence is not excluded (as a sum of translational and relative motion).

Influence of the properties of the plate surface on the oscillations of the cramped drop

We consider free and forced oscillations of a clamped liquid drop. The drop is surrounded by an incompressible fluid of a different density. In equilibrium, the drop has the form of a circular cylinder bounded axially by parallel solid planes, and the contact angle is right. These plates have different surface (chemical, mechanical, and geometrical) properties. The solution is represented as a Fourier series in eigenfunctions of the Laplace operator. The resulting system of complex equations for unknown amplitudes was solved numerically. The fundamental frequency of free oscillations can vanish in a certain interval of values of the Hocking parameter. The length of this interval depends on the aspect ratio of the drop. Frequencies of other eigenmodes of the drop decrease monotonically with increasing Hocking parameters.

DETERMINATION OF TENSIONS OF SUSPENDED ROPES OF THE BRIDGE BY FREQUENCY OF NATURAL OSCILLATIONS

The article deals with the issues of development and use of equipment for digital measurements of rope tension based on the frequency of free transverse oscillations. accurate measurement of rope tension is of practical importance. The relevance of this task for the national economy of Ukraine lies in the need to develop a simple and reliable method and equipment for quick and convenient measurement by construction engineers and operating engineers of rope tension in the construction industry. The practical significance of the work lies in the need to timely bring the span structure of the Arch Bridge of the Podilsky bridge crossing over the Dnipro River in Kyiv to the design position after the installation of metal structures with the transfer of the weight of the bridge truss to four main supports and 64 suspension cables with full unloading of four mounting supports for further arrangement of the bridge deck and completion of construction works. The object of the research is the process of measuring the tension of suspension ropes from the own weight of the metal part of the bridge during bringing the truss of the Arch Bridge to the design position of the metal structural elements of the bridge. The subject of the study is the determination of the impact of the combined movement of the upper attachment points (threaded cylindrical couplings) of 64 suspended ropes on the redistribution of the frequencies of natural oscillations and tensions in their contours.

Time‐ and frequency‐domain analyses of the hydraulic system using a discrete numerical scheme based on equivalent circuit modelling

Mathematical modeling is widely used for the time-and frequency-domain analysis of hydraulic systems. To explore the complicated hydraulic transients of the piping system under various operating conditions, different formulations of the equivalent circuit model (ECM) are systematically discussed in this study. In particular, comparative studies are performed regarding the restrictions on the tempo-spatial discretization of the time-domain ECM and the state-of-the-art numerical schemes. In addition, the eigenanalysis of the free oscillation response and frequency features of the forced oscillation response were discussed in detail, respectively. Furthermore, the influence of spatial discretization on the frequency response precision for ECM is also quantified. Analytical results indicate that the ECM can achieve satisfactory performances in both numerical simulation and frequency analysis with affordable computational effort and relatively loose tempo-spatial discretization restriction.

DETERMINATION OF NATURAL FREQUENCIES AND FORMS OF FREE VIBRATIONS OF THE GRATE OF THE RAW COTTON CLEANER

The article discusses the operation of the grate proposed by the authors with grates made of composite material, with a polyamide matrix and a filler of basalt fibre. Experimental grates are mounted on elastic rubber supports with non-linear rigidity. On the basis of finite element modelling, the analysis of the forms and natural frequencies of free oscillations of the grate of cotton cleaner from large litter was carried out. It is shown that the first 12 natural frequencies are located in a narrow range from 2.579 to 2.692 Hz. This is due to the design features of the grate. It is shown that during operation the grate performs forced oscillations with frequencies 45…75 Hz.

Fano resonance in the essentially nonlinear classic system

The classic analogue of the Fano resonance is considered for the essentially nonlinear systems. The closed expressions for the oscillation amplitudes turn out to be universal in terms of the frequency of free oscillations for the various kinds of the nonlinearity. The conditions of the transition between different branches of the amplitude–frequency relation are formulated in the framework of the concept of the limiting phase trajectory. The problem of the instability of the oscillations that leads to the beating between excited and driven oscillators is considered. The analysis of the non-stationary dynamical equations allows us to define the regions of the instability on the amplitude–frequency plane.