Undamped Oscillations Research Articles

Analysis of the results of the study of vibrations the supporting structure of the gates of spillway structures during the operation of hydroelectric power plants

The article discusses the features of the operation of the main gates of spillway structures during the operation of the HPP. As it is known, the main spillway gates are installed at the outlet of the spillway openings into the downstream of the hydroelectric power station. The gates serve to block the openings of the spillway structure, which is opened to discharge water into a flood, as well as in the event of a disaster with an overlying hydroelectric power plant, thereby ensuring reliable operation of the entire water supply path of the hydroelectric power plant. Using the example of the Zhigulevskaya HPP, during the operation of the gates of the spillway structures, stresses in the gate structures, vibrations of the gate elements and the magnitude of the forces in the gate rod during lifting and lowering operations were determined, which made it possible to identify defects in their operation and determine recommendations for their elimination. In addition, the article discusses the results of the study of vibrations of the supporting structure of the gates of the hydroelectric power plant, which could be obtained only after identifying the features of the main gates on the spillway structures during the operation of the hydropower plant. Regarding the nature and origin of the measured oscillations, it can be said that it is not possible to give a clear explanation of the causes and physical nature of the gate oscillations at the HPP, therefore, the article considers two cases as a result of which periodic undamped gate oscillations may occur. Since it has been established that the periodic undamped oscillations registered on the gate could only be caused by a periodic disturbing force, the cause of such oscillations may be the periodic separation of vortices from the lower face of the gate. As a result, the analysis of the study of vibrations, the supporting structure of the shutter showed that the main disturbing forces for the shutter could be: pulsation of the flow pressure on the upper and lower surfaces of the shutter; dynamic forces arising when the vortices are detached from the lower face of the shutter when water flows out from under it. The article provides recommendations for avoiding vibration of the shutter in subsequent periods of operation.

Self-pulsing and dual-mode lasing in a square microcavity semiconductor laser.

Self-pulsing and dual-mode lasing in a square microcavity semiconductor laser are studied experimentally. Self-sustained pulses originating from undamped relaxation oscillation induced by a two-mode interaction are obtained, as the injection current is slightly above the laser threshold. A repetition frequency of 4.4 GHz and a pulse width of 30-40 ps are obtained at a current of 8 mA. The laser switches to continuous-wave operation when the injection current is higher than a certain value, and dual-mode lasing with 30.7 GHz at 16 mA and 10.7 GHz at 27 mA are observed in the lasing spectra. Furthermore, the relative intensity noise spectra are presented to reveal the relationship between the lasing states and the dynamics induced by relaxation oscillation and mode beating.

COVID-19: From Limit Cycle to Stable Focus

The study aims at investigating a new fundamental property of infectious diseases with natural adaptive immunity that weakens over time—qualitative change (bifurcation) in the behavior of the “virus vs. human” system with an increase in contagiousness. Numerical experiments with a model of the COVID-19 epidemic in Moscow have demonstrated that when the reproduction number R0 is about 4, a qualitative change (bifurcation) occurs in the behavior of the virus–human system. Below this value, the long-term forecast tends toward undamped oscillations; above it, the forecast shows damped oscillations: the amplitudes of epidemic waves decrease gradually, with a constant, very high background level of morbidity that keeps natural immunity near 100%. To confirm this result analytically, we use an original modification of the Euler–Lotka renewal equation, which describes the dynamics of infected patients distributed by disease duration (time since infection) and accounts for immunity. To construct a bifurcation diagram, which illustrates the dependence of the equilibrium stability on the parameter R0, we linearize the equation in the vicinity of the equilibrium point and examine its numerical approximation (discrete form). This approximation can be interpreted as a Leslie model, with the matrix elements dependent on the parameter R0. By examining the roots of the corresponding Lotka polynomial, we can assess the stability of the equilibrium point and verify the basic assumption about the change in the properties of the system with increasing R0—about the transition from undamped oscillations to damped ones. For the bifurcation diagram, we use the functions obtained from the simulation of the COVID-19 epidemic in Moscow. However, observations of the epidemic in other cities and countries support the primary finding of our study regarding the attenuation of epidemic waves.

Slippery rheotaxis: new regimes for guiding wall-bound microswimmers

The near-surface locomotion of microswimmers under the action of background flows has been studied extensively, whereas the intervening effects of complex surface properties remain hitherto unknown. Intending to delineate the shear-driven dynamics near a planar slippery wall, we adopt the squirmer model of microswimmers and employ a three-dimensional analytical-numerical framework in bispherical coordinates. It is interpreted that both the self-propulsion and the external shear flow are redistributed due to hydrodynamic slippage, followed by modulations in the thrust torque on the microswimmer. Phase portraits of the quasi-steady dynamics indicate that the stable upstream swimming states, known as ‘rheotaxis’, are significantly modulated by the slip length compared with the no-slip case. For puller swimmers, an intricate interplay among the modulated interfacial friction near a slippery surface, velocity gradients of the shear flow and the strength of the squirmer parameter promotes a critical shear rate beyond which a wide range of new rheotactic states exist. Consequently, an escaping microswimmer may exhibit rheotaxis or an existing rheotactic state annihilates due to crashing. Although stable states are absent for pushers without steric interactions, transitions from escaping and undamped oscillations to ‘rheotaxis’ occur in the presence of wall repulsion, but only until the other characteristics are overwhelmed by escape due to enhanced shear. Disclosing the ability of hydrodynamic slippage in broadening the scope of migration against a background flow for a wide range of parameters, the present work paves the way for investigations on the entrapment of microswimmers near complex pathways or sorting using selective rheotaxis.

Nonlinear oscillations for beginners. Calculating period using an elementary computational approach

Determining the period of a nonlinear oscillation is a challenging task that requires a strong mathematical background in solving nonlinear differential equations. However, the procedure can be significantly simplified using the area under the 1/|υ|= f(x) graph, where υ is the velocity of the oscillating object and x is its displacement from its equilibrium position. The proposed method requires elementary computational tools and is appropriate for determining the period of any nonlinear undamped oscillation. Characteristic examples are presented, such as the simple pendulum, the oscillation with a power-law restoring force, and the cubic-quintic Duffing oscillator. The proposed approach provides accurate results and is appropriate for introductory physics and mechanics courses.

A modified Krylov–Bogoliubov–Mitropolskii method for solving damped nonlinear oscillators with large oscillation

A modification of the extended Krylov–Bogoliubov–Mitropolskii​ method (by Popov et al.) has been presented to investigate damping effects of strongly nonlinear oscillators when the amplitude of oscillation is large. The modification is similar to the Lindstedt-Poincare method, which was early presented by Cheung et al. for handling un-damped large oscillations. The determination of solution is quite different from earlier all techniques and it is valid whether linear restring force is present or not. Moreover, the solution approaches toward modified Lindstedt-Poincare’s solution when damped force vanishes.

Response of Undamped Oscillators Exposed to Rectangular Pulse Force

Response of Undamped Oscillators Exposed to Rectangular Pulse Force

New constraints on primordial features from the galaxy two-point correlation function

Features in the primordial power spectrum represent the imprinted signal in the density perturbations of the physics and evolution of the early Universe. A measurement of such signals will represents the need to go beyond the minimal assumption made for the initial conditions of the cosmological perturbations. For the first time, we study different templates with undamped oscillations or a bump from the two-point correlation function measured from BOSS DR12 galaxies constraining the amplitude of the features to be at most a few percent. Constraints are competitive to the ones obtained with {\em Planck} DR3.

Mathematical Modeling of Capillary Drawing Stability for Hollow Optical Fibers

The stability problem solution of the manufacturing (drawing) of the quartz capillaries (pipes) for microstructured optical fibers (hole-assisted fiber) is important for determining the effective technological production modes. This importance is also caused by the high cost of fiber production and strict requirements for the accuracy of the fiber’s geometric characteristics. Therefore, a theoretical approach to this problem is relevant and necessary. A modified capillary drawing model that takes into account inertial, viscous, and surface tension forces, as well as all types of heat transfer is proposed in the research. Within the framework of the linear theory of stability, a mathematical model of isothermal and nonisothermal capillary drawing has been developed. The stability of the process is studied depending on the drawing ratio and the Reynolds number. The analysis of the sensitivity of the process to perturbations in the boundary conditions is carried out. The secondary flow that occurs upon transition to the region of instability is also studied. It has been found that at draw ratios above critical values (instability region), undamped oscillations arise. The existence of optimal parameters of the heating element is shown: temperature distribution over the furnace surface and furnace radius, at which the stability of the process of drawing quartz tubes increases significantly (several times).

Global, Non-Scattering Solutions to the Energy Critical Wave Maps Equation

We consider the 1-equivariant energy critical wave maps problem with two-sphere target. Using a method based on matched asymptotic expansions, we construct infinite time relaxation, blow-up, and intermediate types of solutions that have topological degree one. More precisely, for a symbol class of admissible, time-dependent length scales, we construct solutions which can be decomposed as a ground state harmonic map (soliton) re-scaled by an admissible length scale, plus radiation, and small corrections which vanish (in a suitable sense) as time approaches infinity. Our class of admissible length scales includes positive and negative powers of t, with exponents sufficiently small in absolute value. In addition, we obtain solutions with soliton length scale undergoing damped or undamped oscillations in a bounded set, or undergoing unbounded oscillations, for all sufficiently large t.

RESPONSE OF UNDAMPED SDOF SYSTEMS SUBJECTED TO TRIANGULAR PULSE

The single degree of freedom (SDOF) systems have been studied usually via ordinary mathematical tools like calculus, for inquiring about the effects of various pulses like rectangular pulse, step force, triangular pulse, etc., on their responses. In this paper, the SDOF systems such as an undamped mechanical oscillator as well as an undamped electrical oscillator exposed to a triangular pulse have been contemplated and their responses have been determined by the integral Rohit transform (RT). The graphs of responses of SDOF systems versus time have been plotted. It is inferred from the graphs that even though the amplitude of triangular pulse is decreasing linearly, the response of SDOF systems is periodically increasing and decreasing. This paper put forward a new technique for determining the response of SDOF systems (undamped oscillators) exposed to a triangular pulse and reveals that RT is an effective tool for analyzing the SDOF systems.

Quantum dripping of superfluid 4He

Superfluid 4He droplets exhibit robustly quantized dripping periods despite gradually varying input flow rates owing to their high amplitude and undamped oscillation. Since “superfluid” is already placed at the beginning of the sentence, “as a result of superfluidity” may be removed here.

Enhanced evolutionary approach for solving fractional difference recurrent neural network systems: A comprehensive review and state of the art in view of time-scale analysis

<abstract><p>The present research deals with a novel three-dimensional fractional difference neural network model within undamped oscillations. Both the frequency and the amplitude of movements in equilibrium are subsequently estimated mathematically for such structures. According to the stability assessment, the thresholds of the fractional order were determined where bifurcations happen, and an assortment of fluctuations bifurcate within an insignificant equilibrium state. For such discrete fractional-order connections, the parameterized spectrum of undamped resonances is also predicted, and the periodicity and strength of variations are calculated computationally and numerically. Several qualitative techniques, including the Lyapunov exponent, phase depictions, bifurcation illustrations, the $ 0-1 $ analysis and the approximate entropy technique, have been presented with the rigorous analysis. These outcomes indicate that the suggested discrete fractional neural network model has crucial as well as complicated dynamic features that have been affected by the model's variability, both in commensurate and incommensurate cases. Furthermore, the approximation entropy verification and $ \mathbb{C}_{0} $ procedure are used to assess variability and confirm the emergence of chaos. Ultimately, irregular controllers for preserving and synchronizing the suggested framework are highlighted.</p></abstract>

Spatial and temporal dynamics of the emergence of epidemics in the hybrid SIRS+V model of cellular automata

Purpose of this work is to construct a model of infection spread in the form of a lattice of probabilistic cellular automata, which takes into account the inertial nature of infection transmission between individuals. Identification of the relationship between the spatial and temporal dynamics of the model depending on the probability of migration of individuals. Methods. The numerical simulation of stochastic dynamics of the lattice of cellular automata by the Monte Carlo method. Results. A modified SIRS+V model of epidemic spread in the form of a lattice of probabilistic cellular automata is constructed. It differs from standard models by taking into account the inertial nature of the transmission of infection between individuals of the population, which is realized by introducing a "carrier agent" into the model, which viruses act as. The similarity and difference between the dynamics of the cellular automata model and the previously studied mean field model are revealed. Discussion. The model in the form of cellular automata allows us to study the processes of infection spread in the population, including in conditions of spatially heterogeneous distribution of the disease. The latter situation occurs if the probability of migration of individuals is not too high. At the same time, a significant change in the quantitative characteristics of the processes is possible, as well as the emergence of qualitatively new modes, such as the regime of undamped oscillations.

ОБОСНОВАНИЕ КОНСТРУКЦИИ ГЛУБОКОРЫХЛИТЕЛЯ ЧИЗЕЛЬНОГО ТИПА С ОСЦИЛЛЯТОРОМ

Target: using models of interaction of the system of deep–loader deformers with the soil to substantiate the emergence and creation of an innovative system of ripper deformers with elements in the architecture of which oscillatory phenomena occur – flutter. The consideration of the interaction process was based on a critical analysis of various soil models and corresponding schemes of its destruction by horizontal elements of the broken profile of the system of deformers of the deep-digger. The constructed model of interaction is based on the basic principles of mechanics of solids and mechanics of soils (soils). The research was carried out to integrate symmetrically broken soil-forming plates into the architecture of the innovative deep-drilling deformator system in accordance with the current ГОСТ standards and the СТО АИСТ. Data processing was performed using standard and special techniques. A new horizontal element (ground lift) of the system of deformers of the working body of the chisel-type implement, patent for invention No. 2742657, has been developed. The polyline architecture of the soil lift profile allows for better crumbling of the over-compacted soil layer, and also contributes to the occurrence of undamped oscillations. The presence of such an oscillator in the tool deformator system reduces the friction of the soil on the working body, which reduces the energy intensity of the process while improving the quality of loosening. Using the innovative architecture of a chisel deep-loader with an oscillator leads to the occurrence of additional deformations in the decompressed layer and prevents the soil from sticking, as a result of which there is no “soil wedge”, which significantly increases the traction resistance of the deep-loader and reduces the quality of soil loosening. Allows you to profile the loosening area more efficiently and provide the required loosening without turning the treated soil layer to a depth of up to 60 cm.

Investigation of the dynamics of a Capillary Pumped Loop assisted with a mechanical pump: Influence of pump location

Capillary driven two-phase loops are promising solutions for cooling high power electronic devices. A significant upgrade of the operating conditions can be achieved by a small mechanical pump assistance removing the pumping limit inherent to the evaporator porous wick. However, in this couplings, the key issue is to maintain a steady vaporization regime in the evaporator. In the present study, the dynamics of a Capillary Pumped Loop assisted with a centrifugal pump is investigated with both experiments and modeling efforts. The results have revealed (i) that changing the pump location can deteriorate the system robustness with 5 times larger (2.25 kPa) evaporator pressure deviations and (ii) the appearance of a system bifurcation to undamped oscillations at high heat load. A theoretical analysis was performed in frequency domain. It has shown that the loop dynamics is affected by an unexpected transient positive feedback between the pump and the condenser. It has also shown that the loop robustness could be enhanced within a significant range of heat load by a suitable improvement of the condenser efficiency.

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.

Nonlinear dynamics of a semiconductor microcavity laser subject to frequency comb injection.

The nonlinear dynamical behaviors of a semiconductor microcavity laser with frequency comb injection have been experimentally and numerically investigated. The microcavity laser is harmonically locked to a unit fraction of the comb spacing due to the undamped relaxation oscillation at certain conditions, creating additional comb lines with reduced frequency spacing. The stability maps indicating various locking states are obtained based on rate equations, which demonstrates that the locking regions are closely related to the relaxation oscillation. Moreover, the microcavity laser with comb injection leads to spectral broadening of the original comb and the number of comb lines raises from 3 to 13. Owing to the large modulation bandwidth of the microcavity laser, the comb lines and the frequency spacing can be tailored over a wide range by varying the injection parameters.

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.

Closed-Form Solutions to a Forced Damped Rotational Pendulum Oscillator

In this investigation, some analytical solutions to both conserved and non-conserved rotational pendulum systems are reported. The exact solution to the conserved oscillator (unforced, undamped rotational pendulum oscillator), is derived in the form of a Jacobi elliptical function. Moreover, an approximate solution for the conserved case is obtained in the form of a trigonometric function. A comparison between both exact and approximate solutions to the conserved oscillator is examined. Moreover, the analytical approximations to the non-conserved oscillators including the unforced, damped rotational pendulum oscillator and forced, damped rotational pendulum oscillator are obtained. Furthermore, all mentioned oscillators (conserved and non-conserved oscillators) are linearized, and their exact solutions are derived. In addition, all obtained approximations are compared with the four-order Runge–Kutta (RK4) numerical approximations and with the exact solutions to the linearized oscillators. The obtained results can help several authors for discussing and interpreting their results.