Rational Frequency Ratios Research Articles

Hybrid finite-amplitude periodic modes for two uniformly magnetized spheres.

We analyze a system of two uniformly magnetized spheres, one fixed and the other free to slide in frictionless contact with the surface of the first. The centers of the two magnets, and their magnetic moments, are restricted to a plane. We search for sets of initial conditions that yield finite-amplitude oscillatory periodic solutions. We extend two small-amplitude base modes, one with orbital and spin motions that are in phase and the other out of phase, to finite amplitudes and show that the motion for arbitrary oscillatory solutions can be considered to be a nonlinear superposition of these base modes. Some solutions are pure periodic finite-amplitude extensions of one base mode, while others are hybrid finite-amplitude superpositions of the two modes. Hybrid modes with rational frequency ratios are periodic and come in families defined by their frequency ratios. We further characterize hybrid periodic modes by identifying two symmetry classes that describe their relative phases. We see continuous transitions between one finite-amplitude base mode and the other, with one mode gradually transforming into the other. We also calculate frequency spectra of nonperiodic modes, show that the two base modes have well-defined frequencies even for nonperiodic states, and show that periodic solutions can give clues about the behavior of nearby nonperiodic solutions. In the limit of small amplitudes, we confirm that the computed frequencies of these modes agree with small-amplitude analytical results. We also generate a Lyapunov exponent heatmap that reflects periodic and nonperiodic regions of state space.

On Two-Frequency Oscillations of a DC Electric Drive with Pulse Control

Purpose of research is of the paper is to analyze bifurcations of two-frequency oscillations of a DC electric drive with pulse-width control.Methods. The research is based on the construction of a stroboscopic Poincare map, the calculation of saddle periodic orbits and their stable and unstable invariant manifolds.Results. The study of the mechanisms of the occurrence of two-frequency oscillations from a periodic motion that loses stability in a DC electric drive with pulse-width control was carried out. A non-local saddle-node bifurcation leading to resonance (synchronization) on a torus characterized by a pair of independent frequencies when their ratio becomes a rational number, was studied.Conclusion. A bifurcation analysis of the control system of a DC electric drive, the dynamics of which is described by non-smooth nonautonomous differential equations, was carried out. The research was conducted on an iterable map obtained from the specified vector field in an analytical form. It is shown that the system under consideration demonstrates two-frequency oscillations that occur through the Neimark-Sacker bifurcation. In the phase space of the discrete model, a closed invariant curve corresponds to oscillations with two independent frequencies. It is shown that if these frequencies are correlated multiply, then a resonance occurs when the dynamics becomes periodic. But at the same time, the closed curve remains invariant, and the limit points of the orbit form a pair of periodic cycles – stable and saddle, corresponding to a rational frequency ratio. A closed invariant curve is formed by unstable manifolds of a saddle cycle. If the frequency ratio is irrational, then the dynamics is quasi-periodic. The orbits of such motion fill the closed curve everywhere densely.

The final NO-WEAR state due to dual-mode fretting: Numerical prediction and experimental validation

We study fretting wear due to superimposed oscillations in the normal and tangential directions with respect to the contact plane (dual mode fretting wear). In the limit of infinite time, the profile of the indenter tends to a limiting shape, which does not change further. For axisymmetric profiles, the limiting shape is found analytically for the general case of two different frequencies and amplitudes of oscillations in normal and tangential direction. The dependence of worn volume on the frequency ratio is strongly singular showing sharp minima for small rational ratios of frequencies. For the special case of coinciding frequencies, the fretting process was studied both analytically and experimentally. Comparison of experimental results with theoretical predictions showed a good qualitative and quantitative agreement (discrepancy of the order of experimental noise).

Symmetry algebra of the planar anisotropic quantum harmonic oscillator with rational ratio of frequencies

The symmetry algebra of the two-dimensional quantum harmonic oscillator with rational ratio of frequencies is identified as a non-linear extension of the u(2) algebra. The finite dimensional representation modules of this algebra are studied and the energy eigenvalues are de- termined using algebraic methods of general applicability to quantum superintegrable systems.

Nonlinear extension of the u(3) algebra as the symmetry algebra of the three-dimensional anisotropic quantum harmonic oscillator with rational ratios of frequencies and the Nilsson model

The symmetry algebra of the N-dimensional anisotropic quantum har- monic oscillator with rational ratios of frequencies is constructed by a method of general applicability to quantum superintegrable systems. The special case of the 3-dim oscillator is studied in more detail, because of its relevance in the description of superdeformed nuclei and nuclear and atomic clusters. In this case the symmetry algebra turns out to be a nonlinear extension of the u(3) algebra. A generalized angular momentum operator useful for labeling the degenerate states is constructed, clarifying the connection of the present formalism to the Nilsson model.

Nilsson orbitals in quantum superintegrable systems

Recent studies [1] of the valence proton-neutron interaction through certain double differences of binding energies suggest that Nilsson orbitals with ΔK[ΔNΔn_zΔΛ] = 0[110] are responsible for large overlaps of wave functions in heavy deformed nuclei. We show that pairs of orbitals of this kind belong to neighboring irreducible representations of appropriate deformed U(3) algebras [2], being interconnected by deformed creation and annihilation operators. These deformed U(3) algebras correspond to oscillators with rational ratios of frequencies, which are examples of quantum superintegrable systems [3].

Determination of eddy current losses and hysteresis losses in magnetic circuits of electrical machines

The division of the total core losses in the electrical steel of the magnetic circuit into two components – losses dueto hysteresis and eddy currents – is a serious technical problem, the solution of which will effectively design and construct electrical machines with magnetic circuits having low magnetic losses. In this regard, an important parameter is the exponent α, with which the frequency of magnetization reversal is included in the total losses in steel. Theoretically, this indicator can take values from 1 to 2. Most authors take α equal to 1.3, which corresponds to the special case when the eddy current losses are three times higher than the hysteresis losses. In fact, for modern electrical steels, the opposite is true. To refine the index α, an attempt was made to separate the total core losses on the basis that the hysteresis component is proportional to the first degree of the magnetization reversal frequency, and the eddy current component is proportional to the second degree. In the article, the calculation formulas of these components are obtained, containing the values of the total losses measured in idling experiments at two different frequencies, and the ratio of these frequencies. It is shown that the rational frequency ratio is within 1.2. Presented the graphs and expressions to determine the exponent α depending on the measured no-load losses and the frequency of magnetization reversal.

Symmetries of Anisotropic Harmonic Oscillators with Rational Ratios of Frequencies and their Relations to U(N) and O(N+1)

The concept of bisection of a harmonic oscillator or hydrogen atom, vised in the past in establishing the connection between U(3) and 0(4), is generalized into multisection (trisection, tetrasection, etc). It is then shown that all symmetries of the N-dimensional anisotropic harmonic oscillator with rational ratios of frequencies (RHO), some of which are underlying the structure of superdeformed and hyperdeformed nuclei, can be obtained from the U(N) symmetry of the corresponding isotropic oscillator with an appropriate combination of multisections. Furthermore, it is seen that bisections of the N-dimensional hydrogen atom, which possesses an 0(N+1) symmetry, lead to the U(N) symmetry, so that further multisections of the hydrogen atom lead to the symmetries of the N-dim RHO. The opposite is in general not true, i.e. multisections of U(N) do not lead to 0(N+1) symmetries, the only exception being the occurence of 0(4) after the bisection of U(3).

The 1:1 resonance in Hamiltonian systems

Two-degree-of-freedom Hamiltonian systems with an elliptic equilibrium at the origin are characterised by the frequencies of the linearisation. Considering the frequencies as parameters, the system undergoes a bifurcation when the frequencies pass through a resonance. These bifurcations are well understood for most resonances k:l, but not the semisimple cases 1:1 and 1:−1. A two-degree-of-freedom Hamiltonian system can be approximated to any order by an integrable normal form. The reason is that the normal form of a Hamiltonian system has an additional integral due to the normal form symmetry. The latter is intimately related to the ratio of the frequencies. For a rational frequency ratio this leads to S1-symmetric systems. The question we wish to address is about the co-dimension of such a system in 1:1 resonance with respect to left-right-equivalence, where the right action is S1-equivariant. The result is a co-dimension five unfolding of the central singularity. Two of the unfolding parameters are moduli and the remaining non-modal parameters are the ones found in the linear unfolding of this system.

Lissajous trajectories in electromagnetically driven vortices

An experimental and theoretical study of laminar vortical flows driven by oscillating electromagnetic forces that act in orthogonal directions in a shallow electrolytic fluid layer is presented. Forces are generated by the interaction of the field of a dipolar permanent magnet and two imposed alternating electric currents perpendicular to each other with independent frequencies varying in the range of 10–30 mHz. Velocity fields of the time-dependent flow are obtained using particle image velocimetry, while particle tracking allows exploration of the Lagrangian trajectories and time maps. An approximate two-dimensional analytical solution is obtained for the laminar creeping regime so that Lagrangian trajectories are integrated explicitly. These trajectories resemble Lissajous figures with the usual property that, when the ratio of the frequencies of the imposed currents is rational, closed paths are found, while non-closed paths occur when this ratio is irrational. Deviations of this regime that account for slight increase of inertial effects are explored through a quasi-two-dimensional numerical simulation. In this case, non-closed paths are found even for rational frequency ratios. This case was observed in the experiment. Lagrangian trajectories calculated numerically show a qualitative agreement with experimental particle tracking. Furthermore, numerical time maps obtained for increasing inertial effects and rational frequency ratios reveal a chaotic behaviour. Some features of the Lagrangian trajectories are validated experimentally. In particular, topological properties of the calculated and observed time maps are in qualitative agreement. In a characteristic case, a partial time map calculated numerically is compared with the section acquired from the experimental tracking of one particle.

Bi-Hamiltonian structure of the bi-dimensional superintegrable nonlinear isotonic oscillator

The higher-order superintegrability of the two-dimensional isotonic oscillator (noncentral oscillator with inversely quadratic nonlinearities also known as caged anisotropic oscillator) with rational ratio of frequencies is directly related with the existence of some complex functions with interesting Poisson bracket properties. First the properties of these functions are studied and then it is proved that these complex functions determine the existence of a bi-Hamiltonian complex structure. In the second part several real symplectic structures are obtained and the properties of the recursion operators are studied.

Rotational Resonance in milli-tesla fields detected by Field Cycling NMR

Rotational Resonance (R2) between different spin Zeeman levels in samples of adamantane C10H16 (homonuclear R2) and a mixture of C10H16 and C10D16 (both homonuclear and heteronuclear R2) has been studied. A Field Cycling NMR instrument was used to match the external field frequency ν0 to a fixed frequency of sample rotation νr at νr=40, 50 or 60kHz. Rotational Resonance is observed at rational frequency ratios of ν0/νr, such as 12, 23, 32 and 1. The method may prove to become a useful tool for the determination of spin–spin distances in condensed matter.

Observable signature of a background deviating from the Kerr metric

By detecting gravitational wave signals from extreme mass ratio inspiraling sources (EMRIs) we will be given the opportunity to check our theoretical expectations regarding the nature of supermassive bodies that inhabit the central regions of galaxies. We have explored some qualitatively new features that a perturbed Kerr metric induces in its geodesic orbits. Since a generic perturbed Kerr metric does not possess all the special symmetries of a Kerr metric, the geodesic equations in the former case are described by a slightly nonintegrable Hamiltonian system. According to the Poincar\'e-Birkhoff theorem, this causes the appearance of the so-called Birkhoff chains of islands on the corresponding surfaces of section in between the anticipated KAM curves of the integrable Kerr case, whenever the intrinsic frequencies of the system are at resonance. The chains of islands are characterized by finite width, i.e. there is a finite range of initial conditions that correspond to a particular resonance and consequently to a constant rational ratio of intrinsic frequencies. Thus while the EMRI changes adiabatically by radiating energy and angular momentum, by monitoring the frequencies of a signal we can look for a transient pattern, in the form of a plateau, in the evolution of their ratio. We have shown that such a plateau is anticipated to be apparent in a quite large fraction of possible orbital characteristics if the central gravitating source is not a Kerr black hole. Moreover, the plateau in the ratio of frequencies is expected to be more prominent at specific rational values that correspond to the strongest resonances. This gives a possible observational detection of such non-Kerr exotic objects.

A new proof of the higher-order superintegrability of a noncentral oscillator with inversely quadratic nonlinearities

The superintegrability of a rational harmonic oscillator (noncentral harmonic oscillator with rational ratio of frequencies) with nonlinear “centrifugal” terms is studied. In the first part, the system is directly studied in the Euclidean plane; the existence of higher-order superintegrability (integrals of motion of higher order than 2 in the momenta) is proved by introducing a deformation in the quadratic complex equation of the linear system. The constants of motion of the nonlinear system are explicitly obtained. In the second part, the inverse problem is analyzed in the general case of n degrees of freedom; starting with a general Hamiltonian H and introducing appropriate conditions for obtaining superintegrability, the particular “centrifugal” nonlinearities are obtained.

Contrasting the Cartesian and polar forms of the shedding-induced force vector in response to 12 subharmonic and superharmonic mechanical excitations

We mechanically excited the near-field shedding of the von Kármán vortex street of a stationary cylinder at Reynolds number 300 by moving it sinusoidally with 12 different frequencies. We first solved the incompressible Navier–Stokes equations using the finite difference method and obtained unexcited shedding, which occurs at the Strouhal frequency. Then, we applied six subharmonic excitations with rational frequency ratios (defined as the excitation frequency to the Strouhal frequency) in the form 1/n and six superharmonic excitations with frequency ratios n, where n is an integer taking values between 2 and 7. We compared the behavior of the shedding-induced force vector acting on the cylinder in each excitation case to their counterparts in the other excitation cases and to the case of the stationary cylinder. This force vector has two representations: lift and drag (Cartesian) or magnitude and direction (polar). Whereas the first representation is commonly used in theoretical fluid mechanics and engineering applications, we found that the second representation is favored in terms of revealing how the force vector responds to the applied excitations. For example, the mean and standard deviation of the drag do not show clear trends with the excitation frequency, whereas the force magnitude changes quadratically with it. We showed that the theoretical analysis of a simplified nonlinear wake model does in fact support this behavior. We found three modes of shedding-induced force in response to mechanical excitation, depending on the ratio of excitation frequency to Strouhal frequency. Near-field shedding is not entrained by excitation for all subharmonic excitations, but it is entrained for all superharmonic excitations. For entrained shedding, we found two behaviors of the shedding-induced force vector, which is periodic only at odd frequency ratios.

Transition between Kelvin’s equilibria

A transition between Kelvin's equilibrium states is investigated. Using nonlinear theory, we have shown that the transition of polygonal patterns of the hollow vortex core from mode N=2 through N=4 occurs in two steps: quasiperiodicity and frequency locking. We have also shown that this transition can be modeled by a one-dimensional circle map. We extrapolate the present result and hypothesize that the transition between Kelvin's equilibria follows the same route and the ratios of locking frequencies form a Farey sum and staircase function against the control parameter, where the staircase corresponds to the rational frequency ratio, (N-1)/N.

Superintegrability of the caged anisotropic oscillator

We study the caged anisotropic harmonic oscillator, which is a new example of a superintegrable or accidentally degenerate Hamiltonian. The potential is that of the harmonic oscillator with rational frequency ratio (l:m:n) but additionally with barrier terms describing repulsive forces from the principal planes. This confines the classical motion to a sector bounded by the principal planes, or a cage. In three degrees of freedom, there are five isolating integrals of motion, ensuring that all bound trajectories are closed and strictly periodic. Three of the integrals are quadratic in the momenta, the remaining two are polynomials of order 2(l+m−1) and 2(l+n−1). In the quantum problem, the eigenstates are multiply degenerate, exhibiting l2m2n2 copies of the fundamental pattern of the symmetry group SU(3).

Non-symplectic symmetries and bi-Hamiltonian structures of the rational harmonic oscillator

The existence of bi-Hamiltonian structures for the rational Harmonic Oscillator (non-central harmonic oscillator with rational ratio of frequencies) is analyzed by making use of the geometric theory of symmetries. We prove that these additional structures are a consequence of the existence of dynamical symmetries of non-symplectic (non-canonical) type. The associated recursion operators are also obtained.

Commensurate harmonic oscillators: Classical symmetries

The symmetry properties of a classical N-dimensional harmonic oscillator with rational frequency ratios are studied from a global point of view. A commensurate oscillator possesses the same number of globally defined constants of motion as an isotropic oscillator. In both cases invariant phase-space functions form the algebra su(N) with respect to the Poisson bracket. In the isotropic case, the phase-space flows generated by the invariants can be integrated globally to a set of finite transformations isomorphic to the group SU(N). For a commensurate oscillator, however, the group SU(N) of symmetry transformations is found to exist only on a reduced phase space, due to unavoidable singularities of the flow in the full phase space. It is therefore crucial to distinguish carefully between local and global definitions of symmetry transformations in phase space. This result solves the longstanding problem of which symmetry to associate with a commensurate harmonic oscillator.

Sensitivity at the degenerate points of energy levels in a quantum system with nonintegrable perturbation.

The behavior of quantum states and observables of a quasi-integrable quantum system is studied. Sensitivity of the quantum states and nonperiodicity of the average value of observables at the degenerate points of energy levels are observed. The behavior demonstrates the quantum properties corresponding to the classical resonant tori with rational frequency ratio, where the global structure is easily destroyed by a small perturbation. Our result is helpful for further understanding the avoided crossing in hard chaotic quantum systems.