Frequencies Of Small Oscillations Research Articles

On Periodic Motions of a Nearly Autonomous Hamiltonian System in the Occurrence of Double Parametric Resonance

Motions of a 2π-periodic nearly autonomous Hamiltonian system with two degrees of freedom in the neighborhood of the equilibrium position are examined. It is assumed that the Hamiltonian of the system depends on three parameters, namely, e, α, and β, and the system is an autonomous system if e = 0. Let a double parametric resonance, i.e., a situation when one of the frequencies of small linear oscillations of the system in the neighborhood of the equilibrium position is an integer number and the other one is a half-integer number, takes place in an unperturbed (e = 0) system for some α and β values. For sufficiently small, but nonzero, e values in a small neighborhood of the resonance point considered with a fixed resonant value of one parameter (β), the issue of the existence, bifurcations, and stability in the linear approximation of periodic motions of the system is resolved. In the occurrence of multiple resonances of the type under study, periodic motions of a dynamically symmetrical satellite in the neighborhood of its stationary rotation (cylindrical precession) in a slighty elliptical orbit are constructed and linear and nonlinear analyses of their stability are carried out.

The Stability of Relative Equilibrium Positions of a Pendulum on a Mobile Platform

The motion of a pendulum mounted on a platform that rotates around a vertical line with a constant velocity and simultaneously makes specified harmonic oscillations along this vertical line is considered. The platform angular velocity is assumed to be precisely equal to the frequency of small oscillations of the pendulum in the case of an immobile platform. There are relative equilibrium positions of the pendulum when it is oriented along the platform rotation axis (in the hanging or inverted state). The nonlinear problem of stability of these relative equilibria is considered.

The effects of deformation inertia (kinetic energy) in the orbital and spin evolution of close-in bodies

The purpose of this work is to evaluate the effect of deformation inertia on tide dynamics, particularly within the context of the tide response equations proposed independently by Boue et al. (Celest Mech Dyn Astron 126:31–60, 2016) and Ragazzo and Ruiz (Celest Mech Dyn Astron 128(1):19–59, 2017). The singular limit as the inertia tends to zero is analyzed, and equations for the small inertia regime are proposed. The analysis of Love numbers shows that, independently of the rheology, deformation inertia can be neglected if the tide-forcing frequency is much smaller than the frequency of small oscillations of an ideal body made of a perfect (inviscid) fluid with the same inertial and gravitational properties of the original body. Finally, numerical integration of the full set of equations, which couples tide, spin and orbit, is used to evaluate the effect of inertia on the overall motion. The results are consistent with those obtained from the Love number analysis. The conclusion is that, from the point of view of orbital evolution of celestial bodies, deformation inertia can be safely neglected. (Exceptions may occur when a higher-order harmonic of the tide forcing has a high amplitude.)

Two-dimensional motion of a rolling non-circular cylinder on a flat horizontal surface

The two dimensional motion of a generally non-circular non-uniform cylinder on a flat horizontal surface is investigated. Assuming that the cylinder does not slip, energy conservation is used to study the motion in general. Points of returns, and small oscillations around equilibrium configuration are studied. As examples, cylinders are studied for which the cross section is an ellipse, with the center of mass at the center of the ellipse or at a focal point, and the frequencies of small oscillations around their equilibrium configurations are found. The conditions for losing contact or sliding are also investigated. Finally, the motion is studied in more detail for the case of a nearly circular cylinder.

Horizontal Cellular Oscillations Caused by Time-Periodic Resonant Thermal Forcing in Weakly Nonlinear Darcy-Bénard Convection

The onset of Rayleigh-Bénard convection in a horizontally unbounded saturated porous medium is considered. Particular attention is given to the stability of weakly nonlinear convection between two plane horizontal surfaces heated from below. The primary aim is to study the effects on postcritical convection of having small amplitude time-periodic resonant thermal forcing. Amplitude equations are derived using a weakly nonlinear theory and they are solved in order to understand how the flow evolves with changes in the Darcy-Rayleigh number and the forcing frequency. When convection is stationary in space, it is found to consist of one of two different types depending on its location in parameter space: either a convection pattern where each cell rotates in the same way for all time with a periodic variation in amplitude (Type I) or a pattern where each cell changes direction twice within each forcing period (Type II). Asymptotic analyses are also performed (i) to understand the transition between convection of types I and II; (ii) for large oscillation frequencies and (iii) for small oscillation frequencies. In a large part of parameter space the preferred pattern of convection when the layer is unbounded horizontally is then shown to be one where the cells oscillate horizontally—this is a novel form of pattern selection for Darcy-Bénard convection.

Escape of a harmonically forced particle from an infinite-range potential well: a transient resonance

The paper considers a process of escape of classical particle from a one-dimensional potential well by virtue of an external harmonic forcing. We address a particular model of the infinite-range potential well that allows independent adjustment of the well depth and of the frequency of small oscillations. The problem can be conveniently reformulated in terms of action-angle variables. Further averaging provides a nontrivial conservation law for the slow flow. Thus, one can consider the problem in terms of averaged dynamics on primary 1:1 resonance manifold. This simplification allows efficient analytic exploration of the escape process, and yields a theoretical prediction for minimal forcing amplitude required for the escape, as a function of the excitation frequency. This function exhibits a single minimum for certain intermediate frequency value. Numeric simulations are in complete qualitative and reasonable quantitative agreement with the theoretical predictions.

Three-dimensionality of the bulk electronic structure in WTe2

We use temperature- and field-dependent resistivity measurements [Shubnikov--de Haas (SdH) quantum oscillations] and ultrahigh resolution, tunable, vacuum ultraviolet (VUV) laser-based angle-resolved photoemission spectroscopy (ARPES) to study the three-dimensionality (3D) of the bulk electronic structure in WTe2, a type-II Weyl semimetal. The bulk Fermi surface (FS) consists of two pairs of electron pockets and two pairs of hole pockets along the X-Gamma-X direction as detected by using an incident photon energy of 6.7 eV, which is consistent with the previously reported data. However, if using an incident photon energy of 6.36 eV, another pair of tiny electron pockets is detected on both sides of the Gamma point, which is in agreement with the small quantum oscillation frequency peak observed in the magnetoresistance. Therefore, the bulk, 3D FS consists of three pairs of electron pockets and two pairs of hole pockets in total. With the ability of fine tuning the incident photon energy, we demonstrate the strong three-dimensionality of the bulk electronic structure in WTe2. The combination of resistivity and ARPES measurements reveal the complete, and consistent, picture of the bulk electronic structure of this material.

Variational approach to studying solitary waves in the nonlinear Schrödinger equation with complex potentials.

We discuss the behavior of solitary wave solutions of the nonlinear Schrödinger equation (NLSE) as they interact with complex potentials, using a four-parameter variational approximation based on a dissipation functional formulation of the dynamics. We concentrate on spatially periodic potentials with the periods of the real and imaginary part being either the same or different. Our results for the time evolution of the collective coordinates of our variational ansatz are in good agreement with direct numerical simulation of the NLSE. We compare our method with a collective coordinate approach of Kominis and give examples where the two methods give qualitatively different answers. In our variational approach, we are able to give analytic results for the small oscillation frequency of the solitary wave oscillating parameters which agree with the numerical solution of the collective coordinate equations. We also verify that instabilities set in when the slope dp(t)/dv(t) becomes negative when plotted parametrically as a function of time, where p(t) is the momentum of the solitary wave and v(t) the velocity.

Investigation of oscillations of a soliton of a bose condensate of atoms in a trap in the limit of small oscillations of its walls

The motion of the center of a soliton in a trap with oscillating walls is studied analytically and numerically for the case in which the intrinsic frequency of small soliton oscillations in the equilibrium state considerably exceeds the frequency of wall oscillations. this problem can be solved either by applying the gross–pitaevskii equation, which most exactly describes the behavior of the soliton in the trap, or by using the approximate, “mechanical,” equation of motion of the newtonian type for the center of the soliton. an approximate analytical solution of the mechanical equation is obtained and is compared with the numerical solution of the newton equation, while the latter solution is compared with the numerical solution of the gross–pitaevskii equation. good agreement between the first two solutions is revealed. it is also shown that there is a range of parameters in which the numerical solutions of the newton and gross–pitaevskii equations are closest to each other. the frequency-sweeping effect of soliton center oscillations is revealed. an approximate analytical formula for the limiting frequency of these oscillations is obtained and the numerical analysis of this phenomenon is performed.

Influence of block sequence and molecular weight on morphological, rheological and dielectric properties of weakly and strongly segregated styrene-isoprene triblock copolymers

Abstract In this work, morphological, thermal, viscoelastic and dielectric properties of triblock copolymers consisting of styrene (S) and isoprene (I) blocks are discussed. SIS and ISI triblock copolymers were synthesized by sequential anionic polymerization, three of them exhibiting molecular weights below the entanglement molecular weight Me, three of them exhibiting molecular weights in the order of Me and two of them exhibiting molecular weights above Me. The objective of this work is the investigation of the influence of molecular weight and block sequence on relaxation of the block copolymer chains. Morphological investigations using small angle X-ray scattering and transmission electron microscopy studies reveal that a larger molecular weight is associated with a more pronounced microphase separation. The presence of two glass transition temperatures reveals microphase separation of the PI and PS blocks. The Fox-Flory equation was applied in order to describe the molecular weight dependence of the glass transition temperature of the polyisoprene and the polystyrene blocks. The analysis of rheological data reveals a Maxwell fluid behavior for the weakly segregated block copolymers, whereas for the strongly segregated block copolymers a pronounced elastic behavior at low frequencies of small amplitude shear oscillations was observed. In the intermediate regime of molecular weight, a complex viscoelastic behavior appears. The plateau modulus G N o is influenced by the sequence of the PS and the PI blocks (SIS or ISI). Our analysis of segmental and normal mode relaxation in dielectric spectroscopy experiments show that the relaxation processes are strongly influenced by the block sequence (PI blocks tethered at one or both ends) and the molecular weight. As a result, the block sequence in triblock copolymers influences dynamical properties in the glassy state and in the melt.

Dynamics of dissipative self-assembly of particles interacting through oscillatory forces.

Dissipative self-assembly is the formation of ordered structures far from equilibrium, which continuously uptake energy and dissipate it into the environment. Due to its dynamical nature, dissipative self-assembly can lead to new phenomena and possibilities of self-organization that are unavailable to equilibrium systems. Understanding the dynamics of dissipative self-assembly is required in order to direct the assembly to structures of interest. In the present work, Brownian dynamics simulations and analytical theory were used to study the dynamics of self-assembly of a mixture of particles coated with weak acids and bases under continuous oscillations of the pH. The pH of the system modulates the charge of the particles and, therefore, the interparticle forces oscillate in time. This system produces a variety of self-assembled structures, including colloidal molecules, fibers and different types of crystalline lattices. The most important conclusions of our study are: (i) in the limit of fast oscillations, the whole dynamics (and not only those at the non-equilibrium steady state) of a system of particles interacting through time-oscillating interparticle forces can be described by an effective potential that is the time average of the time-dependent potential over one oscillation period; (ii) the oscillation period is critical to determine the order of the system. In some cases the order is favored by very fast oscillations while in others small oscillation frequencies increase the order. In the latter case, it is shown that slow oscillations remove kinetic traps and, thus, allow the system to evolve towards the most stable non-equilibrium steady state.

Oscillometric—Volumetric Measurements of Pure Gas Adsorption Equilibria Devoid of the Non-Adsorption of Helium Hypothesis

A new method is presented to measure pure gas-adsorption equilibria on porous solids or powders without using a hypothesis on the void volume of the solid adsorbents, such as the non-adsorption of helium hypothesis. The proposed method involves combined volumetric/manometric and dynamic measurements, namely, observations of the frequency of small adiabatic oscillations of the adsorptive gas in equilibrium with the adsorbed phase or adsorbate. The oscillations of the adsorptive gas are initiated by small oscillations of a sphere or a cylinder positioned in a vertical tube above the vessel containing the gas and adsorbents [reversion of experiment by Ruchardt (1929), Flammersfeld (1972), Keller (2014)]. Experiments show that adsorbates have two different phases consisting, respectively, of molecules which are only weakly bound to adsorbent’s atoms so that they can participate in the low-frequency gas oscillations (<10 Hz) and other molecules that are strongly bound to adsorbent’s atoms so they are ‘stiff’, ...

Localized states and their stability in an anharmonic medium with a nonlinear defect

A comprehensive analysis of soliton states localized near a plane defect (a defect layer) possessing nonlinear properties is carried out within a quasiclassical approach for different signs of nonlinearity of the medium and different characters of interaction of elementary excitations of the medium with the defect. A quantum interpretation is given to these nonlinear localized modes as a bound state of a large number of elementary excitations. The domains of existence of such states are determined, and their properties are analyzed as a function of the character of interaction of elementary excitations between each other and with the defect. A full analysis of the stability of all the localized states with respect to small perturbations of amplitude and phase is carried out analytically, and the frequency of small oscillations of the state localized on the defect is determined.

Quasi-harmonic oscillatory motion of charged particles around a Schwarzschild black hole immersed in a uniform magnetic field

In order to test the role of large-scale magnetic fields in quasi-periodic oscillation phenomena observed in microquasars, we study the oscillatory motion of charged particles in the vicinity of a Schwarzschild black hole immersed into an external asymptotically uniform magnetic field. We determine the fundamental frequencies of small harmonic oscillations of charged test particles around stable circular orbits in the equatorial plane of a magnetized black hole, and discuss the radial profiles of frequencies of the radial and latitudinal harmonic oscillations in dependence on the mass of the black hole and the strength of the magnetic field. We demonstrate that assuming relevance of resonant phenomena of the radial and latitudinal oscillations of charged particles at their frequency ratio , the oscillatory frequencies of charged particles can be well related to the frequencies of the twin high-frequency quasi-periodic oscillations observed in the microquasars GRS 1915+105, XTE 1550–564 and GRO 1655–40.

Adjacent Equilibria in Highly Flexible Upright Loop on Rigid Foundation

For very slender structural components, self-weight may compete with elastic flexural stiffness in determining equilibrium configurations. In cases where the inherent elastic stiffness is low (relative to self-weight) we observe a variety of types of highly nonlinear behavior in the equilibrium shapes, together with changes in the natural frequencies of small oscillations about these equilibrium configurations. This technical note describes a specific phenomenon observed in experiments on very slender polycarbonate loops. In addition to profound changes in equilibrium shapes as a function of weight-to-stiffness ratio, under some circumstances it is possible to have two adjacent, co-existing equilibrium configurations. This robust, highly nonlinear snap-through behavior is demonstrated by perturbing from one shape to the other.

Evidence for a small hole pocket in the Fermi surface of underdoped YBa2Cu3Oy.

In underdoped cuprate superconductors, the Fermi surface undergoes a reconstruction that produces a small electron pocket, but whether there is another, as yet, undetected portion to the Fermi surface is unknown. Establishing the complete topology of the Fermi surface is key to identifying the mechanism responsible for its reconstruction. Here we report evidence for a second Fermi pocket in underdoped YBa2Cu3Oy, detected as a small quantum oscillation frequency in the thermoelectric response and in the c-axis resistance. The field-angle dependence of the frequency shows that it is a distinct Fermi surface, and the normal-state thermopower requires it to be a hole pocket. A Fermi surface consisting of one electron pocket and two hole pockets with the measured areas and masses is consistent with a Fermi-surface reconstruction by the charge–density–wave order observed in YBa2Cu3Oy, provided other parts of the reconstructed Fermi surface are removed by a separate mechanism, possibly the pseudogap.

Taylor dispersion in oscillatory flow in rectangular channels

This paper focuses on exploring the effect of the side walls on dispersion in oscillatory Poiseuille flows in rectangular channels. The method of multiple time scales with regular expansions is utilized to obtain analytical expressions for the effective dispersivity D3D⁎. The dispersion coefficient is of the form D3D⁎/Pe2=f(Ω≡ωh2/D,Sc≡D/ν,χ≡w/h) where Pe≡<u>h/D, <u> is the root mean square of the cross-section averaged velocity, ω is the angular velocity, 2w and 2h are, respectively, the width and the height of the cross-section, D is the solute diffusivity, and ν is the fluid kinematic viscosity. The analytical results are compared with full numerical simulations and asymptotic expressions. Also effect of various parameters on dispersion coefficient is explored. For small oscillation frequency Ω the dispersion coefficient approaches the time averaged dispersion of the Poiseuille flow and for large Ω, D3D⁎ scales as Pe2/Ω2 where Pe=<u>h/D. Due to its relative simplicity, the 2D model is frequently utilized for calculating dispersion in channels. However at small dimensionless frequencies the 2D model can significantly underestimate the dispersion, particularly for channels with large χ. At large Ω the dispersion coefficient predicted from the 2D model becomes reasonably accurate, particularly for channels with large χ. For a square channel, the 2D prediction is reasonably accurate for all frequencies. The results of this study will enhance our understanding of transport in microscale systems that are subjected to oscillating flows, and potentially aid technological advances in diverse areas relevant to microfluidic devices.

Complex Dynamics of Spring-Block Earthquake Model Under Periodic Parameter Perturbations

A simple model of earthquake nucleation that may account for the onset of chaotic dynamics is proposed and analyzed. It represents a generalization of the Burridge–Knopoff single-block model with Dieterich–Ruina's rate- and state-dependent friction law. It is demonstrated that deterministic chaos may emerge when some of the parameters are assumed to undergo small oscillations about their equilibrium values. Implementing the standard numerical methods from the theory of dynamical systems, the analysis is carried out for the cases having one or two periodically variable parameters, such that the appropriate bifurcation diagrams, phase portraits, power spectra, and the Lyapunov exponents are obtained. The results of analysis indicate two different scenarios to chaos. On one side, the Ruelle–Takens–Newhouse route to chaos is observed for the cases of limit amplitude perturbations. On the other side, when the angular frequency is assumed constant for the value near the periodic motion of the block in an unperturbed case, variation of oscillation amplitudes probably gives rise to global bifurcations, with immediate occurrence of chaotic behavior. Further analysis shows that chaotic behavior emerges only for small oscillation frequencies and higher perturbation amplitudes when two perturbed parameters are brought into play. If higher oscillation frequencies are assumed, no bifurcation occurs, and the system under study exhibits only the periodic motion. In contrast to the previous research, the onset of chaos is observed for much smaller values of the stress ratio parameter. In other words, even the relatively small perturbations of the control parameters could lead to deterministic chaos and, thus, to instabilities and earthquakes.

Global stability of flow in symmetric wavy channels

AbstractA two-dimensional global stability analysis is numerically conducted for the basic fully developed steady flow inside symmetric wavy channels. The relative amplitude of channel modulation, defined as ratio of wall modulation amplitude to mean hydraulic diameter, is fixed via the analysis at a large value of 0.15. The relative channel wavelength, defined as the ratio of wall modulation wavelength to mean hydraulic diameter, is varied between 1 and 5. An important feature of the present approach is the detailed consideration of the streamwise conditions imposed on the flow, which allows a tailored restriction of the possible disturbances by modifying the number of channel sections that set the periodicity. Stability of the base flow is determined on the basis of the spectral structure analysis conducted, possible after calculation of the complete eigenvalue spectrum and facilitated by the spectral method employed. Two different destabilization mechanisms have been identified for the geometry studied. For small relative channel wavelengths, with values below approximately 2.7, the flow is destabilized via a Hopf bifurcation produced by a Tollmien–Schlichting wave at Reynolds numbers in the approximated range from 265 to 324. For large relative channel wavelengths, with values above approximately 2.5, a pitchfork bifurcation (or a Hopf bifurcation with very small oscillation frequency), not previously reported in the literature and produced by a symmetric stationary disturbance, is found at Reynolds numbers in the approximated range from 146 to 230.

Dynamics of current-carrying string loops in the Kerr naked-singularity and black-hole spacetimes

Current-carrying string-loop dynamics is studied in the Kerr spacetimes. With attention concentrated to the axisymmetric motion of string loops around the symmetry axis of both black-hole (BH) and naked singularity (NS) spacetimes, it is shown that the resulting motion is governed by the presence of an outer tension barrier and an inner angular momentum barrier that are influenced by the BH or NS spin. We classify the string dynamics according to properties of the energy boundary function (effective potential) for the string loop motion. We have found that for NS there exist new types of energy boundary function, namely those with off-equatorial minima. Conversion of the energy of the string oscillations to the energy of the linear translational motion has been studied. Such a transmutation effect is much more efficient in the NS spacetimes because of lack of the event horizon. For BH spacetimes efficiency of the transmutation effect is only weakly spin dependent. Transition from the regular to chaotic regime of the string-loop dynamics is examined and used for explanation of the string-loop motion focusing problem. Radial and vertical frequencies of small oscillations of string loops near minima of the effective potential in the equatorial plane are given. These can be related to high-frequency quasiperiodic oscillations observed near black holes.