Free Oscillations Research Articles

«МЕХАНИКАЛЫҚ ТЕРБЕЛІСТЕР» ТАРАУЫН ОҚЫТУ ӘДІСТЕМЕСІ

Taimuratova Lidiya Ungarbaevna Candidate of Physical and Mathematical Sciences, Acting Associate Professor, Sh.Yessenov Caspian state university of technology and engineering, Aktau, Kazakhstan Almagambetova Aigul Aldazharovna Candidate of Pedagogical Sciences, senior lecturer, Korkyt Ata Kyzylorda University, Kyzylorda, Kazakhstan Zhubanyshov Akzhaidar 1st-year master's student, MPPHYS-22, Sh.Yessenov Caspian state university of technology and engineering, Aktau, Kazakhstan METHODS OF TEACHING THE CHAPTER «MECHANICAL VIBRATIONS» Abstract. This article talks about mechanical vibrations. Mechanical vibrations occur in nature, in physical phenomena. Mechanical vibrations in most cases occur in all areas of physics. The physical nature of vibrations can be different, therefore mechanical, electromagnetic, etc. vibrations are distinguished. However, they are characterized by the same characteristics and the same equations for different oscillatory processes. Hence the expediency of a unified approach to the study of vibrations of various physical nature. Depending on the nature of the impact on the oscillatory system, various are distinguished. Since the subject of physics begins with the seventh grade, it is necessary to repeat laws and laws and phenomena in it, and they are in close connection with each other. The article deals with spring vibrations, after mentioning pendulums, about the formation of the concept of an oscillatory system. Interprets a mathematical pendulum as a standard for explaining free oscillations. When explaining the oscillations of the pendulum, the formulas should be described using a vector. Then they say go to projections. Oscillatory movements are common in our daily life. For example, the movement of leaves during the wind, when a metal plate bends and falls on one side, its fall on a swing, such as a swing, a cradle, a pendulum of a wall clock, a load suspended on a spring, a dombra threshold, a reference to a scale can all be examples of swings. This chapter is very difficult to explain, so it will be necessary to give examples of phenomena occurring in everyday life, nature, take into account the enthusiasm of students and explain through physical phenomena. The simplest way to explain the movement of a mathematical pendulum is that if we take into account the air resistance of free oscillation, then at the bottom it, that is, the oscillations, stops. It should be said that in order for the oscillation to occur, it is necessary to influence the force from the outside. Interpreting mechanical vibrations, electromagnetic vibrations should be considered as analogs. When considering mechanical movements, the main attention should be paid to learning through various mathematical transformations.. When teaching the chapter on mechanical vibrations, it is advisable to demonstrate the types of pendulums using a demonstration experiment. Keywords: mechanical vibrations, oscillatory processes, pendulums, springs, waves, loads, wavelengths, frequencies, chemical phenomena, forces, acceleration, acceleration of gravity.

Giant injection-locking bandwidth of a self-pulsing limit-cycle in an optomechanical cavity

Locking of oscillators to ultra-stable external sources is of paramount importance for improving close-to-carrier phase noise in free running oscillators. In most of them, such as Micro-Electro-Mechanical-Systems or LC circuit-based oscillators, the locking frequency range is limited by the robustness of their natural frequency, which comes explicitly related with intrinsic parameters of the system. In this work we report the synchronization of an optically-driven self-pulsing limit-cycle taking place in a silicon optomechanical crystal cavity to an external harmonic signal that modulates the driving laser. Because of the extreme ductility of the natural self-pulsing frequency (several tens of MHz), the injection-locking mechanism is highly efficient and displays giant relative bandwidths exceeding 60%. The external modulation reveals itself as a knob to explore dynamical attractors that are otherwise elusive and, in particular, as a means to initialize a mechanical resonator into a state of self-sustained oscillations driven by radiation pressure forces. Moreover, we exploit the large anharmonicity of the studied limit-cycle to induce injection-locking to integer multiples and fractions of the frequency of the external reference, which can be used for frequency conversion purposes in nano-electro-opto-mechanical systems.

Resonant Oscillations within Multiple Moonpools for a Fixed Rectangular Vessel

_ We present experimental results of resonant free surface oscillations within three circular moonpools arranged in tandem at forward, central, and aft positions of a fixed rectangular vessel in head waves. The piston mode resonance frequency is primarily captured, which decreases with the increase in the vessel draft. The aim is to study the effect of body diffraction on the free surface amplitude and phase of the oscillating water columns at the three locations. The results indicate that, in general, the forward moonpool has the highest response amplitude, whereas the relative amplitudes of the central and aft moonpools depend on the wave frequency. It is observed that the nondimensional response amplitude increases nonlinearly with decreasing wave steepness close to the resonance frequency, while the effect diminishes at lower wave frequencies. The oscillation phase differences between the moonpools show effects of wave-body interaction, a phenomenon dependent on the vessel draft and wave frequency. Finally, the study includes a comparison of the responses at the three moonpool locations between multiple and single configurations. Introduction A moonpool is a vertical opening through the ship deck and open to the sea at the bottom, which is installed in vessels specialized in certain offshore operations. Resonant water column oscillations are encountered in moonpools (Aalbers 1984) due to vessel operations in waves. On the other hand, oscillating water columns (OWCs) have been extensively researched, primarily due to their potential for ocean wave energy conversion (Evans 1978; Heath 2012; Falcão & Henriques 2016). Now, considering design perspectives, the focus of the studies on water column resonance in waves depends on the specific marine application. For example, wave energy converters would require maximized OWC responses for efficient energy capture (Evans & Porter 1995; Morris-Thomas et al. 2007), while large free surface oscillations within moonpools of drillships have adverse effects on the vessel dynamics (Fakuda 1977).

МЕХАНИЧЕСКИЙ ОСЦИЛЛЯТОР БЕЗ НАКОПИТЕЛЯ ПОТЕНЦИАЛЬНОЙ ЭНЕРГИИ

In a monoreactive harmonic oscillator, inert elements can make free sinusoidal oscillations, which are accompanied by the transformation of one inert element kinetic energy into the kinetic energy of another inert element. In this condition the energy of the first inert element is zero. At the same time, the energy of the second element has the maximum value. At the next moment of time, the first element acquires acceleration due to the kinetic energy of the second element, the speed of which begins to decrease. In a classical oscillator, free sinusoidal oscillations are accompanied by an exchange of energy between its elements having the opposite reactivity character. In a spring pendulum, the potential energy of an elastic element is transformed into the kinetic energy of an inert element and vice versa. These elements have the opposite character of reactivity. In an electric oscillatory circuit, the energy of the coil magnetic field is transformed into the energy of the condenser electric field and vice versa. These elements also have the opposite character of reactivity. There are also oscillators in which free sinusoidal oscillations are accompanied by the transformation of the kinetic energy of an inert element or the potential energy of an elastic element into the energy of the coil magnetic field or the energy of the capacitor electric field and vice versa. Free sinusoidal oscillations can occur during the mutual transformation of any physical types of energy.This circumstance is the motive to make an oscillator, in which free sinusoidal oscillations are accompanied by the transformation of the kinetic energy of an inert element into the kinetic energy of another inert element. There are no elements with a different reactivity character in such an oscillator. This type of an oscillator is essentially monoreactive.

On the Countering of Free Vibrations by Forcing: Part I—Non-Resonant and Resonant Forcing with Phase Shifts

The question addressed is whether the free oscillations of a continuous system can be suppressed, or at least the total energy reduced, by applying external forces, using as example the linear undamped transverse oscillations of a uniform elastic string. The non-resonant forcing at an applied frequency, distinct from all natural frequencies, does not interact with the normal modes, whose energy is unchanged, and adds the energy of the forced oscillation, thus increasing the total energy, that is the opposite of the result being sought. The resonant forcing at an applied frequency, equal to one of the natural frequencies, leads to an amplitude growing linearly with time, and hence the energy is growing quadratically with time, implying an increase in total energy after a sufficiently long time. A reduction in total energy is possible over a short time, say over the first period of oscillation, by optimizing the forcing. In the case of a concentrated force, by optimizing its magnitude and location, the total energy with forcing in one period is reduced by a modest maximum of 2% relative to the free oscillation alone. The conclusion is similar for several concentrated forces. In the case of a continuously distributed force, by optimizing the spatial distribution, it is possible to reduce the energy of the total oscillation to one-fourth of that of the free oscillation over the first period of vibration. This shows that continuously distributed forces are more effective at vibration suppression than point forces.

Innovative methods and technologies while examining equations of mathematical physics

Teaching disciplines of physics and mathematics in the age of information technology requires innovative teaching methods application. These methods complement traditional forms; they are aimed at developing the systematic and creative thinking of students. The disciplines of the physical and mathematical profile include such discipline as “Equations of Mathematical Physics”. They help students develop skills in creating mathematical models of physical processes and phenomena. It is advisable to apply software packages to create computer models of situational problems when studying the discipline “Equations of Mathematical Physics”. The mathematical packages application makes it possible to take various situations by changing the input parameters of the model on the basis of computer models of physical processes. The problems of mathematical physics are based on the study of the second-order partial differential equations of three main types: hyperbolic, elliptic, and parabolic. The paper discusses the mathematical package MathCad application in the study of free and forced vibrations of a string. Mathematical models of free and forced oscillations are presented. The d’Alembert formula for solving equations of hyperbolic type is presented. The authors describe stages of the study of free vibrations of the membrane in MathCad. Bessel functions are constructed. Membrane surface animation has been created. Observations of membrane vibrations were carried out.

Normal mode observability of radial anisotropy in the Earth’s mantle

SUMMARYObservations of seismic anisotropy provide useful information to infer directions of mantle flow. However, existing global anisotropic tomography models are not consistent, particularly in the lower mantle. Therefore, the interpretation of seismic anisotropy in terms of mantle dynamics and evolution remains difficult. While surface and body waves are commonly used to build radially anisotropic tomography models, they provide heterogeneous data coverage and the radial anisotropy structure retrieved using these data may be biased by the use of imperfect crustal corrections. Normal modes, the free oscillations of the Earth, automatically provide global data coverage and their sensitivity to shear wave (vs) and compressional wave (vp) velocity makes them suitable to study both vs and vp anisotropy in the mantle. In this study, we assess whether current normal mode splitting data have sufficient sensitivity to lower mantle anisotropy to potentially constrain it. We consider the uncertainties in the data and the effect of inaccuracies in crustal thickness corrections and the assumed scaling between vp and vs. We perform forward modelling of normal mode data using six different 3-D global radially anisotropic tomography models to document how strong and widespread anisotropy has to be to be observable in current normal mode data. We find that, on average 50% of the spheroidal and 55% of the toroidal modes investigated show significant sensitivity to vs anisotropy, while roughly 57% of the spheroidal modes also have strong sensitivity to vp anisotropy. Moreover, we find that the normal mode data fit varies substantially for the various anisotropic tomography models considered, with the addition of anisotropy not always improving the data fit. While we find that crustal thickness corrections do not strongly impact modes that are sensitive to the lower mantle, we observe a trade-off between radial anisotropy and vp scaling for these modes. As long as this is taken into consideration, our findings suggest that existing normal mode data sets can provide valuable information on both vs and vp anisotropy in the mantle.

Observation of free oscillations after the 2010 Chile and 2011 Japan earthquakes by superconducting gravimeter in Kutch, Gujarat, India

In this paper, we present observations of free oscillations of the Earth after major earthquakes in Chile (February 27, 2010, MW8.8) and Japan (March 11, 2011, MW9.1) using data from the dual-sphere superconducting gravimeter (SG - 055), installed at Badargadh (23°0.47 N, 70°0.62 E), Kutch, Gujarat, India in March 2009. To see the noise characteristics, we calculated the power spectral density of the gravity time series of 5 quiet days in the frequency band 0.05–20 mHz using the new low noise model (NLNM) as a reference. We compared the noise level of the Badargadh site to other SG sites around the world. This shows that the Badargadh SG is in a low noise state. We find that the noise increases at frequencies below 1 mHz. Such a characteristic is also observed in Djougou (Afrique, Benin) and Strasbourg (France). Using theoretical tides for Gujarat, we estimated a scale factor of about −814 nm/s2/V for Grav1 (lower-sphere) and about −775 nm/s2/V for Grav2 (upper-sphere). We corrected the influence of atmospheric pressure from the one-second gravity data before switching to the frequency domain. We extracted a total of 53 Earth's Free Oscillations (EFO) modes during the earthquake in Japan and about 47 EFO modes during the earthquake in Chile. We are able to extract the lowest 0S2 spheroidal mode (0.30945 mHz or 54 min) and 0S0 radial mode (0.81439 mHz or 20 min). The longer time series shows individual 0S2 singlets and 0S3 (0.46855 mHz) singlets due to the Coriolis splitting effect. We cross-referenced the frequencies of these modes using the PREM model and previous global observations. The correlation coefficient between the observed and the PREM model for these two events are 0.999 for Japan earthquake and 0.993 for Chile earthquake. This validates the quality of the data useful for low-frequency studies in seismology. We also calculated the relative deviations of our observed fundamental modes with previously determined observed and theoretical values. We found that the relative deviations of our observed free oscillations do not exceed 0.5%, indicating good correlations.

Resonant AC-AC Converter With Multi-Power Level Controller for Inductive EV Charging Systems

Design of Inductive Power Transfer (IPT) controller to tune EV charging level in a wide range is of great importance. The IPT power level controller is proposed to regulate the power transfer rate by modulating the active state and zero state counts of the converter. During active states, energy injection is applied to the IPT compensation tank which increases the IPT resonant current; during the zero state; no injection occurs, and the resonant current of IPT tank is in free oscillation. The FPGA-based IPT power controller is designed to be applied on single stage AC-AC matrix converter which allows direct connection to the main grid without any need of DC conversion; the controller is also operational in dual-stage AC-DC-AC topologies. The proposed IPT power controller is self-tuned without any need to wireless communication channel and offers more simplicity, higher speed, ease of implementation and lower cost. The efficiency up to 90.73% is achieved for the proposed IPT controller thanks to soft-switching (ZCS and ZVS). Finally, ModelSim-Quartus simulation results as well as experimental validation results are provided.

Chaotic Dynamics of a Periodically Forced Duffing Oscillator with Cubic Quintic Septic Power Nonlinearities

In this paper, we investigate the chaotic dynamics of Cubic Quintic Septic Duffing oscillator. The conditions of the system parameters for the existence of fixed points and the stability of the undamped free Cubic Quintic Septic Duffing oscillator are obtained. Multiple scales method is employed to determine the various resonance states of the system through frequency-amplitude response curves. Using linear damping, septic coefficient, excitation amplitude and excitation frequency as control parameters, the bifurcation diagrams are drawn numerically and the results are confirmed using phase space trajectories and their corresponding Poincaré sections. The influence of septic nonlinearity, linear damping and excitation amplitude in controlling the chaotic behavior of the system is also investigated.

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.

Tsunami free oscillations excited by potential local earthquake sources in Bohai Sea, China

The Bohai Sea is the most seismic active region offshore mainland China. Its coastal region is highly populated. The present work proposed a comprehensive scheme for identifying and examining free oscillations with limited number of tide gauges in an irregular semi-enclosed basin. We combined modal analysis, spectral analysis, and multiple-scenario simulation to investigate tsunami free oscillations in the Bohai Sea Basin and three local bays including Liaodong Bay, Bohai Bay, and Laizhou Bay. The Bohai Sea had broadband periods of free oscillations ranging from 228 to 30 min. There existed both along-shore and cross-shore oscillations in most eigenmodes. The ultra-long dominant periods above 50 min were attributed to the shallow water depth (<20 m). Some free oscillation modes appeared in both the Bohai Sea and local bay domains. In addition, the spatial patterns of dominant eigenmodes were characterized and validated by performing spectral analysis for an Mw 7.8 tsunami scenario. The energy was mainly concentrated in the first three modes (171, 146, and 93 min). This is the first study to characterize the areas prone to tsunami hazards in the Bohai Sea, and the methodology can be further extended to other coastal regions.

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).

The beating effect in uniaxial oriented polymer materials

Elastic-relaxation properties of uniaxial oriented polymer materials under dynamic deformation mode are investigated. A theoretical explanation of the phenomenon of the occurrence of beats is proposed. Experimental confirmation of the obtained theoretical results is given. Using the barrier theory and applying the balance equation of the number of cluster transitions through the energy barrier, taking into account the transition time, a refined constitutive equation of the viscoelasticity of the polymer materials is obtained. Experimental studies in the dynamic mode were carried out by the method of the longitudinal low-amplitude oscillations. For a polymer material, taking into account the time of transition of the cluster through the energy barrier, a refined constitutive equation is obtained. The resulting equation is a second-order differential equation and admits a periodic solution. The application of the obtained equation to the study of free longitudinal low-amplitude oscillations in uniaxial oriented polymer materials is considered. It is shown that the solution of the equation admits two close complex roots, the existence of which leads to the effect of the observed beats. The relations between the parameters of the oscillatory process and viscoelastic characteristics are obtained. The experiment showed that the polymer materials under study have a complex form of the oscillatory process, similar to the phenomenon of beats, in a certain range of load. The dependence of the tangent of the mechanical losses angle calculated from the main frequency on the stress has an acute maximum in the stress range where beats are observed. According to the obtained theoretical ratios and experimentally determined attenuation coefficient and main frequency, the time of transition of the cluster through the energy barrier or the lifetime of the cluster in this energy state is determined. The use of additional information obtained during experiments in static modes made it possible to determine the relaxation time for this material. The calculation on the example of polyethyleneterephthalate (PET) filament showed that the theory is consistent with the experiment. Based on the experimental data obtained during the study of free longitudinal oscillations and the solution of the refined constitutive equation, it is possible to determine the necessary viscoelastic characteristics which makes it possible to predict dynamic deformation processes in polymer materials.

A New Approach to Formation and Directed Radiation of Powerful Short Radio Pulses

This article details a new method for formation and directional radiation of short high-power electromagnetic (EM) pulses, which is based on a wave analog of the Smith–Purcell effect. The presented device cycles through two stages. At the stage of energy accumulation, a slow wave of an open guiding structure interacts with a periodic chain of storage resonators and excites high-Q free field oscillations in each resonator of this chain. The phase of oscillations inside the storage resonators differs by a constant value, and the amplitude increases with time. At the radiation stage, the chain of storage resonators is opened to radiation (as it usually happens in active compressors of EM pulses), and short powerful pulses burst out into free space in strictly defined directions. The proposed design is considered in detail for the simplest model of a device. The results obtained during its rigorous electrodynamic analysis look rather promising; the continuation of this work can contribute to solving a number of urgent problems of physics and technology of millimeter and submillimeter waves.

Free Oscillations of an Orthotropic Conical Shell

Free Oscillations of an Orthotropic Conical Shell

Electromagnetic analysis of switching cells with dies embedded in printed circuit boards: Application to TAPIR (compacT and modulAr Power modules with IntegRated cooling) technology

• Switching cells. • Free oscillation stray measurements. • Electromagnetic coupling. The TAPIR technology (compacT and modulAr Power modules with IntegRated cooling) requires a multiphysics analysis to prove its interest compared to current power module solutions. To contribute to this objective, this paper focuses on magnetic aspects and, more specifically, on stray inductance evaluation. Through simulation and experimental studies, it is shown that stray inductances close to 1nH can be obtained. Several design aspects are also studied: the presence of inner layers in the PCB and the decoupling capacitors arrangement. Furthermore, the magnetic interactions between switching cells is studied by simulations and experiments. It is shown that the presence of an adjacent cell reduces slightly the stray inductance (max 20%). In the case of paralleled switching cells, it is also demonstrated that an alternated position of DC link heat sinks can induce a 30% reduction of the stray inductance.

Investigation of Water Evaporation Process at Air/Water Interface using Hofmeister Ions.

Evaporation is an interfacial phenomenon in which a water molecule breaks the intermolecular hydrogen (H-) bonds and enters the vapor phase. However, a detailed demonstration of the role of interfacial water structure in the evaporation process is still lacking. Here, we purposefully perturb the H-bonding environment at the air/water interface by introducing kosmotropic (HPO4-2, SO4-2, and CO3-2) and chaotropic ions (NO3- and I-) to determine their influence on the evaporation process. Using time-resolved interferometry on aqueous salt droplets, we found that kosmotropes reduce evaporation, whereas chaotropes accelerate the evaporation process, following the Hofmeister series: HPO4-2 < SO4-2 < CO3-2 < Cl- < NO3- < I-. To extract deeper molecular-level insights into the observed Hofmeister trend in the evaporation rates, we investigated the air/water interface in the presence of ions using surface-specific sum frequency generation (SFG) vibrational spectroscopy. The SFG vibrational spectra reveal the significant impact of ions on the strength of the H-bonding environment and the orientation of free OH oscillators from ∼36.2 to 48.4° at the air/water interface, where both the effects follow the Hofmeister series. It is established that the slow evaporating water molecules experience a strong H-bonding environment with free OH oscillators tilted away from the surface normal in the presence of kosmotropes. In contrast, the fast evaporating water molecules experience a weak H-bonding environment with free OH oscillators tilted toward the surface normal in the presence of chaotropes at the air/water interface. Our experimental outcomes showcase the complex bonding environment of interfacial water molecules and their decisive role in the evaporation process.

Innovative statistical approach on graphical optimization and closed-form dynamic response of the poroelastic nanocomposite sandwich structure

Present research with the aid of multivariate analysis of variance explores the closed-form dynamics of the poroelastic nanocomposite sandwich plate. Additionally, graphical optimization of the system’s vibration by proper adjusting the involved parameters is provided by employing response surface methodology. p-value and F-value tests are executed to determine the contribution percentage of each parameter on the dynamics of the structure. The nanocomposite face-sheets are made of graphene-platelets reinforced polymer. The main relations governing the dynamics of the initially stressed sandwich system are defined in the background of theory known as linear poroelasticity. Bidirectional format of discrete singular convolution approach (2D-DSCA) is employed in order to determine the design-points’ dynamic performance. Validity of the applied solution procedure is tested through comparison with the outcomes of the first-rate articles. In this research, it is revealed that the graphical optimization is a reliable way to determine the proper adjustment of the parameters for achieving the optimum vibration performance. Moreover, it is observed that the aspect ratio of the plate is the most decisive factor in the free oscillation response of the sandwich plate compared with other factors whose influence are examined in the present article including GPL’s weight-fraction, GPL’s dispersion patterns, in-plane initial stresses, and the poroelasticity coefficient.

Misalignment effects on the performance and stability of x-ray free electron laser oscillator

We report on the effects of transverse spatial misalignment on the performance and stability of x-ray free electron laser oscillators (XFELOs). For this study, we adopt the FEL driven paraxial resonator model in which the transverse profile of the radiation field is represented by Gauss-Hermite mode expansion. Then, we apply Gaussian optics for transporting the radiation field along with the misalignment in the optical cavity. We divide the misalignment into relevant categories of static and periodic types to understand their effects on FEL gain, lasing, and saturation. After that, we associate the first order moments of the radiation field to that of driven simple harmonic oscillator models to identify regions of instability. Throughout this report, we validate our models and analyses via theoretical calculations, analytical approximations, and simulations.