Showing papers on "Normal mode published in 2012"

PT-Symmetric Electronics

Abstract: We show both theoretically and experimentally that a pair of inductively coupled active LRC circuits (dimer), one with amplification and another with an equivalent amount of attenuation, display all the features which characterize a wide class of non-Hermitian systems which commute with the joint parity-time operator: typical normal modes, temporal evolution, and scattering processes Utilizing a Liouvilian formulation, we can define an underlying -symmetric Hamiltonian, which provides important insight for understanding the behavior of the system When the -dimer is coupled to transmission lines, the resulting scattering signal reveals novel features which reflect the -symmetry of the scattering target Specifically we show that the device can show two different behaviors simultaneously, an amplifier or an absorber, depending on the direction and phase relation of the interrogating waves Having an exact theory, and due to its relative experimental simplicity, -symmetric electronics offers new insights into the properties of -symmetric systems which are at the forefront of the research in mathematical physics and related fieldsThis article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’

Wave motion and dispersion phenomena: Veering, locking and strong coupling effects

Abstract: The dispersion curves describe wave propagation in a structure, each branch representing a wave mode. As frequency varies the wavenumbers change and a number of dispersion phenomena may occur. This paper characterizes, analyzes, and quantifies these phenomena in general terms and illustrates them with examples. Two classes of phenomena occur. Weak coupling phenomena—veering and locking—arise when branches of the dispersion curves interact. These occur in the vicinity of the frequency at which, for undamped waveguides, the dispersion curves in the uncoupled waveguides would cross: if two dispersion curves (representing either propagating or evanescent waves) come close together as frequency increases then the curves either veer apart or lock together, forming a pair of attenuating oscillatory waves, which may later unlock into a pair of either propagating or evanescent waves. Which phenomenon occurs depends on the product of the gradients of the dispersion curves. The wave mode shapes which describe the deformation of the structure under the passage of a wave change rapidly around this critical frequency. These phenomena also occur in damped systems unless the levels of damping of the uncoupled waveguides are sufficiently different. Other phenomena can be attributed to strong coupling effects, where arbitrarily light stiffness or gyroscopic coupling changes the qualitative nature of the dispersion curves.

Asymptotic Distribution of Quasi-Normal Modes for Kerr–de Sitter Black Holes

Abstract: We establish a Bohr–Sommerfeld type condition for quasi-normal modes of a slowly rotating Kerr–de Sitter black hole, providing their full asymptotic description in any strip of fixed width. In particular, we observe a Zeeman-like splitting of the high multiplicity modes at a = 0 (Schwarzschild–de Sitter), once spherical symmetry is broken. The numerical results presented in Appendix B show that the asymptotics are in fact accurate at very low energies and agree with the numerical results established by other methods in the physics literature. We also prove that solutions of the wave equation can be asymptotically expanded in terms of quasi-normal modes; this confirms the validity of the interpretation of their real parts as frequencies of oscillations, and imaginary parts as decay rates of gravitational waves.

Spatial damping of propagating kink waves due to mode coupling

Abstract: Aims. We investigate the damping process for propagating transverse velocity oscillations, observed to be ubiquitous in the solar corona, due to mode coupling. Methods. We perform 3D numerical simulations of footpoint-driven transverse waves propagating in a low β coronal plasma with a cylindrical density structure. Mode coupling in an inhomogeneous layer leads to the coupling of the kink mode to the Alfven mode, observed as the decay of the transverse kink oscillations. Results. We consider the spatial damping profile and find a Gaussian damping profile of the form exp(−z/Lg) to be the most congruent with our numerical data, rather than the exponential damping profile of the form exp(−z/Ld) used in normal mode analysis. Our results highlight that the nature of the driver itself will have a substantial influence on observed propagating kink waves. Conclusions. Our study suggests that this modified damping profile should be taken into account when using coronal seismology to infer local plasma properties from observed damped oscillations.

Relating normal vibrational modes to local vibrational modes with the help of an adiabatic connection scheme.

Abstract: Information on the electronic structure of a molecule and its chemical bonds is encoded in the molecular normal vibrational modes. However, normal vibrational modes result from a coupling of local vibrational modes, which means that only the latter can provide detailed insight into bonding and other structural features. In this work, it is proven that the adiabatic internal coordinate vibrational modes of Konkoli and Cremer [Int. J. Quantum Chem. 67, 29 (1998)] represent a unique set of local modes that is directly related to the normal vibrational modes. The missing link between these two sets of modes are the compliance constants of Decius, which turn out to be the reciprocals of the local mode force constants of Konkoli and Cremer. Using the compliance constants matrix, the local mode frequencies of any molecule can be converted into its normal mode frequencies with the help of an adiabatic connection scheme that defines the coupling of the local modes in terms of coupling frequencies and reveals how avoided crossings between the local modes lead to changes in the character of the normal modes.

Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink

Abstract: We explore the conservative and dissipative dynamics of a two-degree-of-freedom (2-DoF) system consisting of a linear oscillator and a lightweight nonlinear rotator inertially coupled to it. When the total energy of the system is large enough, the motion of the rotator is, generically, chaotic. Moreover, we show that if the damping of the rotator is sufficiently small and the damping of the linear oscillator is even smaller, then the system passes through a cascade of resonance captures (transient internal resonances) as the total energy gradually decreases. Rather unexpectedly, all these captures have the same principal frequency but correspond to different nonlinear normal modes (NNMs). In each NNM, the rotator is phase-locked into periodic motion with two frequencies. The NNMs differ by the ratio of these frequencies, which is approximately an integer for each NNM. Essentially non-integer ratios lead to incommensurate periods of ‘slow’ and ‘fast’ motions of the rotator and, thus, to its chaotic behavior between successive resonance captures. Furthermore, we show that these cascades of resonance captures lead to targeted energy transfer (TET) from the linear oscillator to the rotator, with the latter serving, in essence, as a nonlinear energy sink (NES). Since the inertially-coupled NES that we consider has no linearized natural frequency, it is capable of engaging in resonance with the linear oscillator over broad frequency and energy ranges. The results presented herein indicate that the proposed rotational NES appears to be a promising design for broadband shock mitigation and vibration energy harvesting.

Kinetic cascade beyond magnetohydrodynamics of solar wind turbulence in two-dimensional hybrid simulations

Abstract: The nature of solar wind turbulence in the dissipation range at scales much smaller than the large magnetohydrodynamic (MHD) scales remains under debate. Here, a two-dimensional model based on the hybrid code abbreviated as A.I.K.E.F. is presented, which treats massive ions as particles obeying the kinetic Vlasov equation and massless electrons as a neutralizing fluid. Up to a certain wavenumber in the MHD regime, the numerical system is initialized by assuming a superposition of isotropic Alfven waves with amplitudes that follow the empirically confirmed spectral law of Kolmogorov. Then, turbulence develops and energy cascades into the dispersive spectral range, where also dissipative effects occur. Under typical solar wind conditions, weak turbulence develops as a superposition of normal modes in the kinetic regime. Spectral analysis in the direction parallel to the background magnetic field reveals a cascade of left-handed Alfven/ion-cyclotron waves up to wave vectors where their resonant absorption se...

Modeling and vibration mode analysis of a ball screw drive

Abstract: Positioning systems for machine tools are generally driven by ball screws due to their high stiffness and low sensitivity to external perturbations. However, as modern machine tools increase their velocity and acceleration of positioning, the resonant modes of these systems could be excited degrading the trajectory tracking accuracy. Therefore, a dynamic model including the vibration modes is required for machine design as well as for controller selection and tuning. This work presents a high-frequency dynamic model of a ball screw drive. The analytical formulation follows a comprehensive approach, where the screw is modeled as a continuous subsystem, using Ritz series approximation to obtain an approximate N-degree-of-freedom model. Based on this model, the axial and angular components of each mode function are studied for different transmission ratios to determine the degree of coupling between them. After that, the frequency variation of each mode was studied for different carriage positions and different moving masses. Finally, an analysis of these results applied to controller design and parameter estimation is also presented.

Feasibility study of a 3D vibration-driven electromagnetic MEMS energy harvester with multiple vibration modes

Abstract: A novel electromagnetic energy harvester (EH) with multiple vibration modes has been developed and characterized using three-dimensional (3D) excitation at different frequencies. The device consists of a movable circular-mass patterned with three sets of double-layer aluminum (Al) coils, a circular-ring system incorporating a magnet and a supporting beam. The 3D dynamic behavior and performance analysis of the device shows that the first vibration mode of 1285 Hz is an out-of-plane motion, while the second and third modes of 1470 and 1550 Hz, respectively, are in-plane at angles of 60 ◦ (240 ◦ ) and 150 ◦ (330 ◦ ) to the horizontal (x-) axis. For an excitation acceleration of 1 g, the maximum power density achieved are 0.444, 0.242 and 0.125 μ Wc m −3 at vibration modes of I, II and III, respectively. The experimental results are in good agreement with the simulation and indicate a good potential in the development of a 3D EH device. (Some figures may appear in colour only in the online journal)

The effects of magnetic-field geometry on longitudinal oscillations of solar prominences

Abstract: We investigate the influence of the geometry of the solar filament magnetic structure on the large-amplitude longitudinal oscillations. A representative filament flux tube is modeled as composed of a cool thread centered in a dipped part with hot coronal regions on either side. We have found the normal modes of the system and establish that the observed longitudinal oscillations are well described with the fundamental mode. For small and intermediate curvature radii and moderate to large density contrast between the prominence and the corona, the main restoring force is the solar gravity. In this full wave description of the oscillation a simple expression for the oscillation frequencies is derived in which the pressure-driven term introduces a small correction. We have also found that the normal modes are almost independent of the geometry of the hot regions of the tube. We conclude that observed large-amplitude longitudinal oscillations are driven by the projected gravity along the flux tubes and are strongly influenced by the curvature of the dips of the magnetic field in which the threads reside.

Sound absorption and transmission through flexible micro-perforated panels backed by an air layer and a thin plate.

Abstract: This paper describes theoretical and experimental investigations into the sound absorption and transmission properties of micro-perforated panels (MPP) backed by an air cavity and a thin plate. A fully coupled modal approach is proposed to calculate the absorption coefficient and the transmission loss of finite-sized micro-perforated panels-cavity-panel (MPPCP) partitions with conservative boundary conditions. It is validated against infinite partition models and experimental data. A practical methodology is proposed using collocated pressure-velocity sensors to evaluate in an anechoic environment the transmission and absorption properties of conventional MPPCPs. Results show under which conditions edge scattering effects should be accounted for at low frequencies. Coupled mode analysis is also performed and analytical approximations are derived from the resonance frequencies and mode shapes of a flexible MPPCP. It is found that the Helmholtz-type resonance frequency is deduced from the one associated to the rigidly backed MPPCP absorber shifted up by the mass-air mass resonance of the flexible non-perforated double-panel. Moreover, it is shown analytically and experimentally that the absorption mechanisms at the resonances are governed by a large air-frame relative velocity over the MPP surface, with either in-phase or out-of-phase relationships, depending on the MPPCP parameters.

Generalized Eigenvalue Analysis of Symmetric Prestressed Structures Using Group Theory

Abstract: As conventional approaches for calculating natural frequencies do not make full use of the inherent symmetry of a structure, the rising degree of freedoms often leads to significant increase in computational demand. In this study, a simplified technique for analyzing dynamic characteristics of symmetric prestressed structures is described using group theory. First, the generalized eigenvalue equation of a prestressed structure based on tangent stiffness matrix and lumped mass matrix is built to get natural frequencies and the corresponding vibration shapes in which the contribution of initial prestresses is considered. A symmetry-adapted coordinate system for the structure is adopted to block-diagonalize the stiffness and mass matrices. The complexity of generalized eigenvalue equation is reduced by solving the mutually independent subspaces, and thus natural frequencies and the corresponding vibration modes could be obtained. Illustrative examples point out the general procedure, and show the sup...

Kinetic cascade beyond magnetohydrodynamics of solar wind turbulence in two-dimensional hybrid simulations

Abstract: The nature of solar wind turbulence in the dissipation range at scales much smaller than the large MHD scales remains under debate. Here a two-dimensional model based on the hybrid code abbreviated as A.I.K.E.F. is presented, which treats massive ions as particles obeying the kinetic Vlasov equation and massless electrons as a neutralizing fluid. Up to a certain wavenumber in the MHD regime, the numerical system is initialized by assuming a superposition of isotropic Alfv\'en waves with amplitudes that follow the empirically confirmed spectral law of Kolmogorov. Then turbulence develops and energy cascades into the dispersive spectral range, where also dissipative effects occur. Under typical solar wind conditions, weak turbulence develops as a superposition of normal modes in the kinetic regime. Spectral analysis in the direction parallel to the background magnetic field reveals a cascade of left-handed Alfv\'en/ion-cyclotron waves up to wave vectors where their resonant absorption sets in, as well as a continuing cascade of right-handed fast-mode and whistler waves. Perpendicular to the background field, a broad turbulent spectrum is found to be built up of fluctuations having a strong compressive component. Ion-Bernstein waves seem to be possible normal modes in this propagation direction for lower driving amplitudes. Also signatures of short-scale pressure-balanced structures (very oblique slow-mode waves) are found.

Normal Mode Determination of Perovskite Crystal Structures with Octahedral Rotations: Theory and Applications

Abstract: Nuclear site analysis methods are used to enumerate the normal modes of $ABX_{3}$ perovskite polymorphs with octahedral rotations. We provide the modes of the fourteen subgroups of the cubic aristotype describing the Glazer octahedral tilt patterns, which are obtained from rotations of the $BX_{6}$ octahedra with different sense and amplitude about high symmetry axes. We tabulate all normal modes of each tilt system and specify the contribution of each atomic species to the mode displacement pattern, elucidating the physical meaning of the symmetry unique modes. We have systematically generated 705 schematic atomic displacement patterns for the normal modes of all 15 (14 rotated + 1 unrotated) Glazer tilt systems. We show through some illustrative examples how to use these tables to identify the octahedral rotations, symmetric breathing, and first-order Jahn-Teller anti-symmetric breathing distortions of the $BX_{6}$ octahedra, and the associated Raman selection rules. We anticipate that these tables and schematics will be useful in understanding the lattice dynamics of bulk perovskites and would serve as reference point in elucidating the atomic origin of a wide range of physical properties in synthetic perovskite thin films and superlattices.

The Raman spectrum of diopside: a comparison between ab initio calculated and experimentally measured frequencies

Abstract: The Raman spectrum of diopside has been calculated by using three purely Density Functional Theory (DFT) Hamiltonians (PBE, WCPBE, LDA), the Hartree-Fock Hamiltonian (HF) and three hybrid HF/DFT ones (B3LYP, WC1LYP, PBE0). A comparison has been done between the calculated frequencies with those measured by Raman spectroscopy on a natural sample, along with several different orientations and beam polarizations, or retrieved from literature; such a comparison demonstrated the excellent performances of the hybrid Hamiltonians in reproducing the vibrational spectrum of the mineral, in line with what it is generally observed in literature concerning other mineral phases. In particular, the mean average absolute discrepancies of the calculated frequencies with respect to the experimental data were: 3.2 (WC1LYP), 4.7 (B3LYP), 6.5 (PBE0), 18.0 (PBE), 9.7 (WCPBE), 7.3 (LDA), and 40.6 cm −1 (HF). The very good quality of the WC1LYP results allowed for a reliable assignment of all of the experimentally observed Raman signals, and the corresponding assignments to specific patterns of atomic vibrational motion (normal modes).

Direct numerical simulation and biglobal stability investigations of the gaseous motion in solid rocket motors

Abstract: In this article, a biglobal stability approach is used in conjunction with direct numerical simulation (DNS) to identify the instability mode coupling that may be responsible for triggering large thrust oscillations in segmented solid rocket motors (SRMs). These motors are idealized as long porous cylinders in which a Taylor–Culick type of motion may be engendered. In addition to the analytically available steady-state solution, a computed mean flow is obtained that is capable of securing all of the boundary conditions in this problem, most notably, the no-slip requirement at the chamber headwall. Two sets of unsteady simulations are performed, static and dynamic, in which the injection velocity at the chamber sidewall is either held fixed or permitted to vary with time. In these runs, both DNS and biglobal stability solutions converge in predicting the same modal dependence on the size of the domain. We find that increasing the chamber length gives rise to less stable eigenmodes. We also realize that introducing an eigenmode whose frequency is sufficiently spaced from the acoustic modes leads to a conventional linear evolution of disturbances that can be accurately predicted by the biglobal stability framework. While undergoing spatial amplification in the streamwise direction, these disturbances will tend to decay as time elapses so long as their temporal growth rate remains negative. By seeding the computations with the real part of a specific eigenfunction, the DNS outcome reproduces not only the imaginary part of the disturbance, but also the circular frequency and temporal growth rate associated with its eigenmode. For radial fluctuations in which the vorticoacoustic wave contribution is negligible in relation to the hydrodynamic stability part, excellent agreement between DNS and biglobal stability predictions is ubiquitously achieved. For axial fluctuations, however, the DNS velocity will match the corresponding stability eigenfunction only when properly augmented by the vorticoacoustic solution for axially travelling waves associated with the Taylor–Culick profile. This analytical approximation of the vorticoacoustic mode is found to be quite accurate, especially when modified using a viscous dissipation function that captures the decaying envelope of the inviscid acoustic wave amplitude. In contrast, pursuant to both static and dynamic test cases, we find that when the frequency of the introduced eigenmode falls close to (or crosses over) an acoustic mode, a nonlinear mechanism is triggered that leads to the emergence of a secondary eigenmode. Unlike the original eigenmode, the latter materializes naturally in the computed flow without being artificially seeded. This natural occurrence may be ascribed to a nonlinear modal interplay in the form of internal, eigenmode-to-eigenmode coupling instead of an external, eigenmode pairing with acoustic modes. As a result of these interactions, large amplitude oscillations are induced.

Networks of nonlinear superconducting transmission line resonators

Abstract: We investigate a network of coupled superconducting transmission line resonators, each of them made nonlinear with a capacitively shunted Josephson junction coupling to the odd flux modes of the resonator. The resulting eigenmode spectrum shows anticrossings between the plasma mode of the shunted junction and the odd resonator modes. Notably, we find that the combined device can inherit the complete nonlinearity of the junction, allowing for a description as a harmonic oscillator with a Kerr nonlinearity. Using a dc SQUID instead of a single junction, the nonlinearity can be tuned between 10 kHz and 4 MHz while maintaining resonance frequencies of a few gigahertz for realistic device parameters. An array of such nonlinear resonators can be considered a scalable superconducting quantum simulator for a Bose–Hubbard Hamiltonian. The device would be capable of accessing the strongly correlated regime and be particularly well suited for investigating quantum many-body dynamics of interacting particles under the influence of drive and dissipation.

Vibration Properties of High-Speed Planetary Gears With Gyroscopic Effects

Abstract: This study investigates the modal property structure of high-speed planetary gears with gyroscopic effects. The vibration modes of these systems are complex-valued and speeddependent. Equally-spaced and diametrically-opposed planet spacing are considered. Three mode types exist, and these are classified as planet, rotational, and translational modes. The properties of each mode type and that these three types are the only possible types are mathematically proven. Reduced eigenvalue problems are determined for each mode type. The eigenvalues for an example high-speed planetary gear are determined over a wide range of carrier speeds. Divergence and flutter instabilities are observed at extremely high speeds. [DOI: 10.1115/1.4006646]

Magnetar oscillations – II. Spectral method

Abstract: The seismological dynamics of magnetars are largely determined by a strong hydromagnetic coupling between the solid crust and the fluid core. In this paper, we set up a ‘spectral’ computational framework in which the magnetar’s motion is decomposed into a series of basis functions that are associated with the crust and core vibrational eigenmodes. A general relativistic formalism is presented for evaluation of the core Alfven modes in the magnetic flux coordinates, as well for eigenmode computation of a strongly magnetized crust of finite thickness. By considering coupling of the crustal modes to the continuum of Alfven modes in the core, we construct a fully relativistic dynamical model of the magnetar which allows: (i) fast and long simulations without numerical dissipation; and (ii) very fine sampling of the stellar structure. We find that the presence of strong magnetic field in the crust results in localizing of some high-frequency crustal elastomagnetic modes with the radial number n≥ 1 to the regions of the crust where the field is nearly horizontal. While the hydromagnetic coupling of these localized modes to the Alfven continuum in the core is reduced, their energy is drained on a time-scale of ≪1 s. Therefore, the puzzle of quasi-periodic oscillations with frequencies larger than 600 Hz still stands.

A geometrically exact approach to the overall dynamics of elastic rotating blades—part 1: linear modal properties

Abstract: A geometrically exact mechanical model for the overall dynamics of elastic isotropic rotating blades is proposed. The mechanical formulation is based on the special Cosserat theory of rods which includes all geometric terms in the kinematics and in the balance laws without any restriction on the geometry of deformation besides the enforcement of the local rigidity of the blade cross sections. All apparent forces acting on the blade moving in a rotating frame are accounted for in exact form. The role of internal kinematic constraints such as the unshearability of the slender blades is discussed. The Taylor expansion of the governing equations obtained via an Updated Lagrangian formulation is then employed to obtain the linearized perturbed form about the prestressed configuration under the centrifugal forces. By applying the Galerkin approach to the linearized equations of motion, the linear eigenvalue problem is solved to yield the frequencies and mode shapes. In particular, the natural frequencies of unshearable blades including coupling between flapping, lagging, axial and torsional components are investigated. The angular speeds at which internal resonances may arise due to specific ratios between the frequencies of different modes are determined thus shedding light onto the overall modal couplings in rotating beam structures depending on the angular speed regime. The companion paper (part 2) discusses the nonlinear modes of vibration away from internal resonances.