Статті в журналах: "Subharmonic mode"

1

IBADULLAEV, Muhtarhan, Akrom N. TOVBAEV, and Azamat Zh ESENBEKOV. "On the General Theory for Analysis of Subharmonic Oscillations in Three-Phase Ferroresonance Circuits and Systems." Elektrichestvo, no. 12 (2021): 35–44. http://dx.doi.org/10.24160/0013-5380-2021-12-35-44.

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The general theory for analysis of subharmonic oscillations at a frequency of ω/3 in three-phase ferroresonance circuits is presented. The occurrence and existence of ferroresonance oscillations at subharmonic frequencies in power transmission lines and power supply systems is highly undesirable, since they cause overvoltages at various frequencies. At the same time, there is an extensive class of nonlinear electrical circuits in which the excitation of autoparametric oscillations at the frequency of subharmonics forms the basis of phase-discrete frequency converting devices serving as secondary power sources. To study the regularities of excitation and maintaining of subharmonic oscillations at a frequency of ω/3 in three-phase ferroresonance circuits, theoretical and experimental studies of an equivalent model of a three-phase circuit with nonlinear inductance were carried out. A generalized nonlinear differential equation for a three-phase circuit with mixed connection of its elements is derived. The steady-state mode of subharmonic oscillations at a frequency of ω/3 is analyzed using the small parameter (averaging) method, which made it possible to determine their existence domains and circuit critical parameters. A mathematical model and algorithm for calculating autoparametric oscillations have been developed to study the subharmonic oscillation excitation processes at a frequency of ω/3 in three-phase ferroresonance circuits depending on the initial conditions, circuit parameters and input voltage. The theoretical study results have been confirmed experimentally.

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2

Monkewitz, Peter A. "Subharmonic resonance, pairing and shredding in the mixing layer." Journal of Fluid Mechanics 188 (March 1988): 223–52. http://dx.doi.org/10.1017/s0022112088000710.

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An instability-wave analysis is presented to describe the spatial evolution of a fundamental mode and its subharmonic on an inviscid parallel mixing layer. It incorporates explicitly the weakly nonlinear interaction between the two modes. The computational finding that the development of the subharmonic, leading eventually to pairing or shredding, crucially depends on its phase relation with the fundamental is fully confirmed. Furthermore it is shown that a critical fundamental amplitude has to be reached before the (spatial) subharmonic becomes phase locked with the fundamental and exhibits a modified growth rate. Then the analysis is exploited to explain the occurrence of amplitude modulations in ‘natural’ mixing layers and to estimate the width of the subharmonic spectral peaks. Also, the case of oblique subharmonic waves is briefly touched upon. In the last part, ways are explored to model non-parallel effects, i.e. to handle the saturation of the rapidly growing subharmonic. Using this wave description, the role of mode interaction in the ‘vortex pairing’ and ‘shredding’ process is assessed.

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3

Eigenwillig, Christoph M., Wolfgang Wieser, Benjamin R. Biedermann, and Robert Huber. "Subharmonic Fourier domain mode locking." Optics Letters 34, no. 6 (March 3, 2009): 725. http://dx.doi.org/10.1364/ol.34.000725.

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4

Panigrahi, P. K., and S. Acharya. "Multi-Modal Forcing of the Turbulent Separated Shear Flow Past a Rib." Journal of Fluids Engineering 126, no. 1 (January 1, 2004): 22–31. http://dx.doi.org/10.1115/1.1637634.

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Experiments have been conducted to study the development of flow behind a surface mounted rib under different phase controlled excitation. Single mode excitation and multi-mode excitation with different relative phases are studied. The results presented include the coherent and random components of the turbulent energy and shear stresses, the energy exchange with the mean flow and between the modes, and the phase decorrelation of the coherent components. The fundamental-subharmonic excitation does not provide any significant improvements in the shear layer growth over the fundamental excitation. The shear layer growth correlates with the subharmonic mode development. The large scale structures are significant even after the reattachment region as evident from the magnitude of the coherent components of the turbulent energy and the shear stress. The binary exchange terms are significant in the near-field region whose sign is phase dependent, i.e., it reverses its sign based on the phase difference between the fundamental and 1st subharmonic mode. The location of the fundamental and subharmonic peaks are different from the peak location of their respective energy exchange with the mean flow; this is attributed to the significance of the binary energy exchange between the fundamental and the subharmonic mode in this region. The excitation regularizes the flow leading to low phase jitter in the near field region. The origin and development of phase decorrelation is attributed primarily to the subharmonic instability.

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5

Yang, Ruixin, Frederic K. Browand, Pierre Coullet, and Patrick Huerre. "A model of defect-induced pairing in mixing layers." Journal of Fluid Mechanics 248 (March 1993): 403–23. http://dx.doi.org/10.1017/s0022112093000825.

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Experiment have shown that the well-known vortex pairing process may take place first in defect regions where the fundamental structure is weakest. A model is introduced here to describe this defect-induced pairing process. The model is constructed in such a way that, in a certain parameter range, a stable fundamental mode and a stable subharmonic mode may coexist. The numerical simulation demonstrates that, when initial conditions consist of a dominant fundamental with one or more defects, the subharmonic component is preferentially generated in the cores of these defects. Moreover, the results also indicate that pairing may first commence wherever the fundamental mode is weakest provided the white-noise level of the subharmonic is high enough. The numerical results are in good agreement with experimental observations.

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6

Friedland, L., J. Fajans, and E. Gilson. "Subharmonic autoresonance of the diocotron mode." Physics of Plasmas 7, no. 5 (May 2000): 1712–18. http://dx.doi.org/10.1063/1.873989.

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7

Sheard, G. J., M. C. Thompson, and K. Hourigan. "Subharmonic mechanism of the mode C instability." Physics of Fluids 17, no. 11 (November 2005): 111702. http://dx.doi.org/10.1063/1.2139682.

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8

GORDEYEV, S. V., and F. O. THOMAS. "Temporal subharmonic amplitude and phase behaviour in a jet shear layer: wavelet analysis and Hamiltonian formulation." Journal of Fluid Mechanics 394 (September 10, 1999): 205–40. http://dx.doi.org/10.1017/s0022112099005650.

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Fourier and wavelet transformation techniques are utilized in a complementary manner in order to characterize temporal aspects of the transition of a planar jet shear layer. The subharmonic is found to exhibit an interesting temporal amplitude and phase variation that has not been previously reported. This takes the form of intermittent π-shifts in subharmonic phase between two fixed phase values. These phase jumps are highly correlated with local minima of the subharmonic amplitude. In contrast, the fundamental amplitude and phase show no such behaviour. The temporal phase behaviour of the subharmonic has the effect of intermittently disrupting the phase lock with the fundamental. A dynamical systems model is developed which is based on a classic vortex representation of the shear layer. The Hamiltonian formulation of the problem is shown to provide remarkable agreement with the experimental results. All the essential aspects of the temporal amplitude and phase behaviour of the subharmonic are reproduced by the model including amplitude-dependent effects. The model is also shown to provide a dynamical systems based explanation for time-averaged amplitude and phase behaviour observed in these as well as earlier experiments. The results of experiments involving both bimodal forcing at fundamental and subharmonic frequencies with prescribed initial effective phase angle as well as experiments involving only fundamental excitation over an amplitude range extending two orders of magnitude are presented. The temporal subharmonic amplitude and phase behaviour is observed in bimodal forcing experiments in those regions of the flow characterized by subharmonic mode suppression and vortex tearing events (even if the forcing amplitudes are quite large). In addition, temporal subharmonic amplitude and phase behaviour is the rule in experiments involving low-amplitude forcing of the fundamental and the natural development of the subharmonic.

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9

CHEN, WEN-YAU, and C. F. CHEN. "Effect of gravity modulation on the stability of convection in a vertical slot." Journal of Fluid Mechanics 395 (September 25, 1999): 327–44. http://dx.doi.org/10.1017/s0022112099006011.

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The nature of instability occurring in a differentially heated infinite slot under steady gravity depends only on the Prandtl number of the contained Boussinesq fluid. For fluids with Pr < 12.5, the instability is shear dominated and onsets in a steady convection mode; for fluids with Pr > 12.5, the instability is buoyancy dominated and onsets in an oscillatory mode. In this paper, we examine the effect of gravity modulation on the stability characteristics of convection in an infinite slot with both kinds of fluids, in particular, Pr = 1 and Pr = 25. Using the method of Sinha & Wu (1991), we are able to obtain accurate results without excessive numerical integration in the linear stability analysis by Floquet theory. Results show that, for Pr = 1, at a non-dimensional oscillation frequency ω = 20, the critical state alternates between the synchronous and subharmonic modes. At higher frequencies, ω > 100, all critical states occur in the synchronous mode. For Pr = 25, with a modulation amplitude ratio of 0.5, resonant interaction occurs in the neighbourhood of ω = 2σc, where σc is the oscillation frequency of the instability at the critical state under steady gravity. This resonant interaction is destabilizing, with the critical Grashof number being reduced by approximately 20% from that at steady gravity. It is due to the presence of a detached subharmonic branch of the marginal stability curve. In frequency ranges where the detached subharmonic branch is absent, the critical state is in the quasi-periodic mode consisting of two waves of different oscillation frequencies whose sum is the forcing frequency. An analysis of the rate of change of the perturbation kinetic energy shows that, for Pr = 1, the instability is shear dominated regardless of the mode of oscillation, synchronous or subharmonic. Similarly, for Pr = 25, the instability is buoyancy dominated whether it is in the quasi-periodic or subharmonic mode. The mode switching is a response to the forcing and is independent of the dominant mechanisms of instability.

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10

Wu, Xuesong, and Philip A. Stewart. "Interaction of phase-locked modes: a new mechanism for the rapid growth of three-dimensional disturbances." Journal of Fluid Mechanics 316 (June 10, 1996): 335–72. http://dx.doi.org/10.1017/s0022112096000572.

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In this paper, we have identified a new mechanism which can promote rapid growth of three-dimensional disturbances. The mechanism involves the interaction between a planar mode and an oblique mode, or a pair of oblique modes, which are phase-locked in the sense that they have the same phase speed. This allows a powerful nonlinear interaction to take place within the common critical layer(s). The disturbance is not required to form a subharmonic resonant triad, and hence the mechanism operates under much less restrictive conditions than does subharmonic resonance (although it is somewhat less powerful). We show that the quadratic interaction between the planar mode and the oblique modes drives an exceptionally large forced mode with the difference frequency, which in turn interacts with the planar mode to contribute the dominant nonlinear effect. This interaction can cause the oblique modes to grow super-exponentially provided that their magnitude is sufficiently small. As a result of the super-exponential growth, the oblique mode may soon become strong enough to produce a feedback effect on the planar mode, so that the interactions eventually become fully coupled. This subsequent stage takes slightly different forms depending on whether a single or a pair of oblique modes is present. Both cases are investigated. Particular attention is paid to symmetric plane shear layers, e.g. a planar wake or jet, for which subharmonic resonance of sinuous modes is inactive.

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11

Li, Fei, and Mujeeb R. Malik. "Fundamental and subharmonic secondary instabilities of Görtler vortices." Journal of Fluid Mechanics 297 (August 25, 1995): 77–100. http://dx.doi.org/10.1017/s0022112095003016.

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The nonlinear development of stationary Görtler vortices leads to a highly distorted mean flow field where the streamwise velocity depends strongly not only on the wall-normal but also on the spanwise coordinates. In this paper, the inviscid instability of this flow field is analysed by solving the two-dimensional eigenvalue problem associated with the governing partial differential equation. It is found that the flow field is subject to the fundamental odd and even (with respect to the Görtler vortex) unstable modes. The odd mode, which was also found by Hall & Horseman (1991), is initially more unstable. However, there exists an even mode which has higher growth rate further downstream. It is shown that the relative significance of these two modes depends upon the Görtler vortex wavelength such that the even mode is stronger for large wavelengths while the odd mode is stronger for short wavelengths. Our analysis also shows the existence of new subharmonic (both odd and even) modes of secondary instability. The nonlinear development of the fundamental secondary instability modes is studied by solving the (viscous) partial differential equations under a parabolizing approximation. The odd mode leads to the well-known sinuous mode of break down while the even mode leads to the horseshoe-type vortex structure. This helps explain experimental observations that Görtler vortices break down sometimes by sinuous motion and sometimes by developing a horseshoe vortex structure. The details of these break down mechanisms are presented.

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12

Wundrow, David W., Lennart S. Hultgren, and M. E. Goldstein. "Interaction of oblique instability waves with a nonlinear plane wave." Journal of Fluid Mechanics 264 (April 10, 1994): 343–72. http://dx.doi.org/10.1017/s0022112094000698.

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This paper is concerned with the downstream evolution of a resonant triad of initially non-interacting linear instability waves in a boundary layer with a weak adverse pressure gradient. The triad consists of a two-dimensional fundamental mode and a pair of equal-amplitude oblique modes that form a subharmonic standing wave in the spanwise direction. The growth rates are small and there is a well-defined common critical layer for these waves. As in Goldstein & Lee (1992), the wave interaction takes place entirely within this critical layer and is initially of the parametric-resonance type. This enhances the spatial growth rate of the subharmonic but does not affect that of the fundamental. However, in contrast to Goldstein & Lee (1992), the initial subharmonic amplitude is assumed to be small enough so that the fundamental can become nonlinear within its own critical layer before it is affected by the subharmonic. The subharmonic evolution is then dominated by the parametric-resonance effects and occurs on a much shorter streamwise scale than that of the fundamental. The subharmonic amplitude continues to increase during this parametric-resonance stage – even as the growth rate of the fundamental approaches zero – and the subharmonic eventually becomes large enough to influence the fundamental which causes both waves to evolve on the same shorter streamwise scale.

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13

Li, Xin-Zheng, Zhan-Guo Bai, Yan Li, and Kun Zhao. "Numerical investigation on oscillatory Turing patterns in a two-layer coupled reaction–diffusion system." Modern Physics Letters B 30, no. 07 (March 20, 2016): 1650085. http://dx.doi.org/10.1142/s0217984916500858.

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In this paper, various kinds of spontaneous dynamic patterns are investigated based on a two-layer nonlinearly coupled Brusselator model. It is found that, when the Hopf mode or supercritical Turing mode respectively plays major role in the short or long wavelength mode layer, the dynamic patterns appear under the action of nonlinearly coupling interactions in the reaction–diffusion system. The stripe pattern can change its symmetrical structure and form other graphics when influenced by small perturbations sourced from other modes. If two supercritical Turing modes are nonlinearly coupled together, the transition from Turing instability to Hopf instability may appear in the short wavelength mode layer, and the twinkling-eye square pattern, traveling and rotating pattern will be obtained in the two subsystems. If Turing mode and subharmonic Turing mode satisfy the three-mode resonance relation, twinkling-eye patterns are generated, and oscillating spots are arranged as square lattice in the two-dimensional space. When the subharmonic Turing mode satisfies the spatio-temporal phase matching condition, the traveling patterns, including the rhombus, hexagon and square patterns are obtained, which presents different moving velocities. It is found that the wave intensity plays an important role in pattern formation and pattern selection.

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14

Sutherland, Bruce R. "Excitation of superharmonics by internal modes in non-uniformly stratified fluid." Journal of Fluid Mechanics 793 (March 16, 2016): 335–52. http://dx.doi.org/10.1017/jfm.2016.108.

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Theory and numerical simulations show that the nonlinear self-interaction of internal modes in non-uniform stratification results in energy being transferred to superharmonic disturbances forced at twice the horizontal wavenumber and frequency of the parent mode. These disturbances are not in themselves a single mode, but a superposition of modes such that the disturbance amplitude is largest where the change in the background buoyancy frequency with depth is largest. Through weakly nonlinear interactions with the parent mode, the disturbances evolve to develop vertical-scale structures that distort and modulate the parent mode. Because pure resonant wave triads do not exist in non-uniformly stratified fluid, parametric subharmonic instability (PSI) is not evident even though noise is superimposed upon the initial state. The results suggest a new mechanism for energy transfer to dissipative scales (from large to small vertical scale and with frequencies larger and smaller than that of the parent mode) through forcing superharmonic rather than subharmonic disturbances.

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15

Pinault, Jean-Louis. "Glaciers and Paleorecords Tell Us How Atmospheric Circulation Changes and Successive Cooling Periods Occurred in the Fennoscandia during the Holocene." Journal of Marine Science and Engineering 9, no. 8 (July 31, 2021): 832. http://dx.doi.org/10.3390/jmse9080832.

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Two major climatic phenomena that occurred during the Holocene are interpreted from the resonance in subharmonic modes of long-period Rossby waves winding around the North Atlantic gyre, the so-called gyral Rossby waves (GRWs). These are, on the one hand, the change in atmospheric circulation that occurred in the North Atlantic in the middle Holocene, and, on the other hand, the occurrence of abrupt cooling events more frequently than what is generally accepted. The amplitude of GRWs is deduced by filtering, within bands characteristic of various subharmonic modes, climate records from the Greenland ice sheet, pollen, and tree rings in northern Fennoscandia, and from two Norwegian glaciers in northern Folgefonna and on the Lyngen peninsula. While the subharmonic modes reflect the acceleration/deceleration phases of the western boundary current, an anharmonic mode is evidenced in the 400–450 year band. Abrupt cooling events of the climate are paced by this anharmonic mode while the western boundary current is decelerating, and the northward heat advection of air favors the melting of the pack ice. Then, the current of the northernmost part of the North Atlantic gyre cools before branching off to the north, which alters its buoyancy. On the other hand, according to high subharmonic modes, high-pressure systems prevailed over the North Atlantic in the first half of the Holocene while low-pressure systems resulted from baroclinic instabilities of the atmosphere dominate during the second half, favoring the growth of glaciers in Scandinavia by a better snowfall in winter and cooler summers.

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16

Wang, Xiaohong, Qisong Huang, Bo Zhang, Di Chen, and Quanxue Guan. "Z-domain modeling of peak current mode control for full-bridge DC-DC buck converters." Journal of Power Electronics 21, no. 1 (October 8, 2020): 27–37. http://dx.doi.org/10.1007/s43236-020-00157-w.

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AbstractTraditional local-averaged state-space modeling for peak current mode (PCM) controls fails to explain the subharmonic oscillation phenomenon when the spectrum is higher than half of the switching frequency. To address this problem, this paper presents a small-signal modeling method in the z-domain, and builds a discrete linear model for the current loop of a full-bridge DC-DC converter. This discrete model is converted into a second-order continuous model that is able to represent the system performance with a wider frequency range. A frequency-domain analysis shows that this model can be used to explain the subharmonic oscillations and unstable characteristics. This provides an engineering guideline for the practical design of slope compensation. The effectiveness of the proposed modeling method has been verified by simulation and experimental results with a prototype working in the Buck mode.

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17

Nikitopoulos, D. E., and J. T. C. Liu. "Nonlinear binary-mode interactions in a developing mixing layer." Journal of Fluid Mechanics 179 (June 1987): 345–70. http://dx.doi.org/10.1017/s0022112087001563.

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In this paper we present the formulation and results of two-wave interactions in a spatially developing shear layer, directed at understanding and interpreting the physical mechanisms that underlie the results of quantitative observation. Our study confirms the existence of Kelly's mechanism that augments the growth of a subharmonic disturbance by extracting energy from its fundamental or vice versa. This mechanism is shown to be strongest in the region where the fundamental begins to return energy to the mean flow and the two wave modes are of comparable energy levels. It is found that the initial conditions and especially the initial phase angle between the two disturbances play a very significant role in the modal development and that of the shear layer itself. A doubling of the shear-layer thickness is shown to take place; the two successive plateaux in its growth are attributed to the peaking in the energy production rates of the fundamental and subharmonic fluctuations.

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18

Hoshida, T., Hai-Feng Liu, M. Tsuchiya, Y. Ogawa, and T. Kamiya. "Subharmonic hybrid mode-locking of a monolithic semiconductor laser." IEEE Journal of Selected Topics in Quantum Electronics 2, no. 3 (1996): 514–22. http://dx.doi.org/10.1109/2944.571752.

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19

Chung, C. A., W. Z. Chien, and Y. H. Hsieh. "Morphological Instabilities in Time Periodic Crystallization." Journal of Mechanics 23, no. 4 (December 2007): 295–302. http://dx.doi.org/10.1017/s1727719100001349.

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AbstractA linear stability analysis is performed on the interface that forms during directional solidification of a dilute binary alloy in the presence of time-periodic growth rates. The basic state, in which the flat crystal-melt interface advances at a steady rate with an oscillation superimposed, is solved analytically by expanding the governing equations in terms of the assumed-small amplitude of modulation. We find that there is a frequency window of stabilization, in which the Mullins-Sekerka instability can be stabilized synchronously. Outside of the window, large input frequencies may destabilize the Mullins-Sekerka mode. The subharmonic mode, which occurs with small wave numbers, is stabilized with increasing the frequency. As for the modulation amplitude, larger amplitude tends to reduce the synchronous mode while enhance the subharmonic mode.

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20

Smith, Gregory D., Charles L. Cox, S. Murray Sherman, and John Rinzel. "Fourier Analysis of Sinusoidally Driven Thalamocortical Relay Neurons and a Minimal Integrate-and-Fire-or-Burst Model." Journal of Neurophysiology 83, no. 1 (January 1, 2000): 588–610. http://dx.doi.org/10.1152/jn.2000.83.1.588.

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We performed intracellular recordings of relay neurons from the lateral geniculate nucleus of a cat thalamic slice preparation. We measured responses during both tonic and burst firing modes to sinusoidal current injection and performed Fourier analysis on these responses. For comparison, we constructed a minimal “integrate-and-fire-or-burst” (IFB) neuron model that reproduces salient features of the relay cell responses. The IFB model is constrained to quantitatively fit our Fourier analysis of experimental relay neuron responses, including: the temporal tuning of the response in both tonic and burst modes, including a finding of low-pass and sometimes broadband behavior of tonic firing and band-pass characteristics during bursting, and the generally greater linearity of tonic compared with burst responses at low frequencies. In tonic mode, both experimental and theoretical responses display a frequency-dependent transition from massively superharmonic spiking to phase-locked superharmonic spiking near 3 Hz, followed by phase-locked subharmonic spiking at higher frequencies. Subharmonic and superharmonic burst responses also were observed experimentally. Characterizing the response properties of the “tuned” IFB model leads to insights regarding the observed stimulus dependence of burst versus tonic response mode in relay neurons. Furthermore the simplicity of the IFB model makes it a candidate for large scale network simulations of thalamic functioning.

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21

Yang, Wei, Hao Wei, and Liang Zhao. "Parametric Subharmonic Instability of the Semidiurnal Internal Tides at the East China Sea Shelf Slope." Journal of Physical Oceanography 50, no. 4 (April 2020): 907–20. http://dx.doi.org/10.1175/jpo-d-19-0163.1.

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AbstractOn the basis of measurements from an observing mooring system, the observational evidence of parametric subharmonic instability (PSI) that transfers energy from semidiurnal internal tides (ITs) to the subharmonic waves at the East China Sea continental shelf slope is presented for the first time. Although the mooring station is very close to the energetic semidiurnal IT generation site, about 76% of the observed shear variance is contained in the near-inertial band, which is found to have comparable upward- and downward-propagating energy components. Bispectra and bicoherence estimates further confirm the occurrence of PSI transferring energy from the low-mode M2 ITs (vertical wavelength of ~1000 m) to high-mode subharmonic waves (vertical wavelength of ~200 m). The calculated energy transfer rate reveals an averaged net value of ~5 × 10−9 W kg−1. Strong temporal variation of is found that is not exactly in phase with the semidiurnal energy flux. After looking into the local vorticity fields, it is strongly suggested that the varying background relative vorticity associated with the evolving Kuroshio has modified the efficiency of PSI at the mooring location through changing the local effective inertial frequency.

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22

SHEARD, G. J., M. C. THOMPSON, K. HOURIGAN, and T. LEWEKE. "The evolution of a subharmonic mode in a vortex street." Journal of Fluid Mechanics 534 (June 21, 2005): 23–38. http://dx.doi.org/10.1017/s0022112005004313.

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23

Sivaramakrishnan, Sudarshan, and Herbert G. Winful. "Subharmonic anti-phase dynamics in coupled mode-locked semiconductor lasers." Optics Letters 42, no. 23 (November 27, 2017): 4905. http://dx.doi.org/10.1364/ol.42.004905.

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24

Hilali, M'F, G. Dewel, and P. Borckmans. "Subharmonic and strong resonances through coupling with a zero mode." Physics Letters A 217, no. 4-5 (July 1996): 263–68. http://dx.doi.org/10.1016/0375-9601(96)00344-1.

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25

Vodopyanov, K. L. "Ultra-broadband mid-infrared frequency combs produced by optical subharmonic generation." Quantum Electronics 52, no. 4 (April 1, 2022): 307–12. http://dx.doi.org/10.1070/qel18012.

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Abstract Optical frequency combs have revolutionised accurate frequency and time measurements and have enabled broadband and simultaneously high resolution spectroscopic measurements that were not previously possible. This paper is an overview of the main results of the previously performed work, describing a new approach to extending frequency combs to the mid-infrared ‘molecular signature’ range using a subharmonic generator based on an optical parametric oscillator operating in degenerate mode. Such an instrument acts as an efficient frequency divider that rigorously down-converts and augments the spectrum of a pump laser frequency comb while maintaining its coherence. Our recent result is the demonstration of a subharmonic system with a two-octave spectrum, 3 – 12 μm, which covers vibrational resonances for most molecular species. Potentially, through frequency division in the coherent subharmonic optical parametric amplifier regime, this method can be used to obtain intense long-wavelength pulses for high-field physics applications.

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26

Spalart, Philippe R., and Kyung-Soo Yang. "Numerical study of ribbon-induced transition in Blasius flow." Journal of Fluid Mechanics 178 (May 1987): 345–65. http://dx.doi.org/10.1017/s0022112087001253.

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The early three-dimensional stages of transition in the Blasius boundary layer are studied by numerical solution of the Navier-Stokes equations. A finite-amplitude two-dimensional wave and low-amplitude three-dimensional random disturbances are introduced. Rapid amplification of the three-dimensional components is observed and leads to transition. For intermediate amplitudes of the two-dimensional wave the breakdown is of subharmonic type, and the dominant spanwise wavenumber increases with the amplitude. For high amplitudes the energy of the fundamental mode is comparable to the energy of the subharmonic mode, but never dominates it; the breakdown is of mixed type. Visualizations, energy histories, and spectra are presented. The sensitivity of the results to various physical and numerical parameters is studied. The agreement with experimental and theoretical results is discussed.

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27

Jurman, L. A., S. E. Deutsch, and M. J. Mccready. "Interfacial mode interactions in horizontal gas—liquid flows." Journal of Fluid Mechanics 238 (May 1992): 187–219. http://dx.doi.org/10.1017/s002211209200168x.

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The behaviour of shear-generated interfacial waves in a cocurrent gas-liquid flow in a small rectangular channel is studied experimentally at conditions close to neutral stability. It is found that the linearly most unstable mode, which typically has a frequency of 8–10 Hz and a wavelength 1–4 cm, grows initially — followed immediately by the first overtone. Measurements of the bicoherence spectrum indicate that the overtone and fundamental are coherent in phase, which suggests that energy is transferred from the fundamental to the linearly stable first overtone. This energy transfer mechanism can stabilize the system, as evidenced by data, which shows that the first mode can saturate with a wave slope small as as 0.005. Theory based on weakly nonlinear mode-interaction equations suggests that this steady state should be stable at conditions close to neutral stability where only overtone modes are present. However, under more severe conditions, where the amplitude of the fundamental mode becomes sufficiently large, a subharmonic mode may be excited. The generation of the subharmonic, when it is linearly stable with respect to the flat film base state, can be interpreted as a linear instability of the steady state containing the fundamental and overtones. Modes that are sidebands (with wavenumbers = k ± δk) to the main peak may also occur. These can participate in interactions with low-frequency modes (i.e. δk) and thereby transfer energy to frequencies much below the fundamental. It is expected that all of these interactions play important roles in determining the wave spectrum of conditions far away from neutral stability.

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28

YOUNG, W. R., Y. K. TSANG, and N. J. BALMFORTH. "Near-inertial parametric subharmonic instability." Journal of Fluid Mechanics 607 (June 30, 2008): 25–49. http://dx.doi.org/10.1017/s0022112008001742.

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New analytic estimates of the rate at which parametric subharmonic instability (PSI) transfers energy to high-vertical-wavenumber near-inertial oscillations are presented. These results are obtained by a heuristic argument which provides insight into the physical mechanism of PSI, and also by a systematic application of the method of multiple time scales to the Boussinesq equations linearized about a ‘pump wave’ whose frequency is close to twice the inertial frequency. The multiple-scale approach yields an amplitude equation describing how the 2f0-pump energizes a vertical continuum of near-inertial oscillations. The amplitude equation is solved using two models for the 2f0-pump: (i) an infinite plane internal wave in a medium with uniform buoyancy frequency; (ii) a vertical mode one internal tidal wavetrain in a realistically stratified and bounded ocean. In case (i) analytic expressions for the growth rate of PSI are obtained and validated by a successful comparison with numerical solutions of the full Boussinesq equations. In case (ii), numerical solutions of the amplitude equation indicate that the near-inertial disturbances generated by PSI are concentrated below the base of the mixed layer where the velocity of the pump wave train is largest. Based on these examples we conclude that the e-folding time of PSI in oceanic conditions is of the order of ten days or less.

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29

Qiu, Zhan, Wenhao Xu, Jiaxin Yuan, and Fuxin Wang. "Secondary resonances in aeroelastic response of oscillating airfoil under dynamic stall." Journal of Vibration and Control 24, no. 23 (March 26, 2018): 5665–80. http://dx.doi.org/10.1177/1077546318764191.

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To clarify the mechanism of the complex aeroelastic responses of flexible blades of helicopter rotors under dynamic stall, experiments on a 2D aeroelastic system are performed. In the spectra of the response from experiment results, special frequency components are found. Then, a numerical method based on the same aeroelastic model is introduced. Here, the flow field is solved using a zonal solver based on vorticity dynamics. When changing a system’s natural frequency, the same extra frequency components in the response spectra are found when particular ratios of natural and forcing frequencies are achieved. Secondary resonances are believed to then happen, which feature a larger response amplitude, multiple periodic motion and a subharmonic peak of driving frequency in the load spectra. With an analysis of the flow field, the 1/2 subharmonic in the airload spectra (i.e. the period doubling of the loads) is believed to be associated with the nonlinear variation of vortex structures. With a dynamic mode decomposition analysis, a counter-rotating vortex interaction instability is detected as the physical mechanism of period doubling. The coincidence of natural frequency with the odd times of the subharmonic leads to the secondary resonances.

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30

Sun, Oliver M., and Robert Pinkel. "Subharmonic Energy Transfer from the Semidiurnal Internal Tide to Near-Diurnal Motions over Kaena Ridge, Hawaii." Journal of Physical Oceanography 43, no. 4 (April 1, 2013): 766–89. http://dx.doi.org/10.1175/jpo-d-12-0141.1.

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Анотація:

Abstract Nonlinear energy transfers from the semidiurnal internal tide to high-mode, near-diurnal motions are documented near Kaena Ridge, Hawaii, an energetic generation site for the baroclinic tide. Data were collected aboard the Research Floating Instrument Platform (FLIP) over a 35-day period during the fall of 2002, as part of the Hawaii Ocean Mixing Experiment (HOME) Nearfield program. Energy transfer terms for a PSI resonant interaction at midlatitude are identified and compared to those for near-inertial PSI close to the M2 critical latitude. Bispectral techniques are used to demonstrate significant energy transfers in the Nearfield, between the low-mode M2 internal tide and subharmonic waves with frequencies near M2/2 and vertical wavelengths of O(120 m). A novel prefilter is used to test the PSI wavenumber resonance condition, which requires the subharmonic waves to propagate in opposite vertical directions. Depth–time maps of the interactions, formed by directly estimating the energy transfer terms, show that energy is transferred predominantly from the tide to subharmonic waves, but numerous reverse energy transfers are also found. A net forward energy transfer rate of 2 × 10−9 W kg−1 is found below 400 m. The suggestion is that the HOME observations of energy transfer from the tide to subharmonic waves represent a first step in the open-ocean energy cascade. Observed PSI transfer rates could account for a small but significant fraction of the turbulent dissipation of the tide within 60 km of Kaena Ridge. Further extrapolation suggests that integrated PSI energy transfers equatorward of the M2 critical latitude may be comparable to PSI energy transfers previously observed near 28.8°N.

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31

Chang, You Min, Junsu Lee, and Ju Han Lee. "A Q-switched, mode-locked fiber laser employing subharmonic cavity modulation." Optics Express 19, no. 27 (December 14, 2011): 26627. http://dx.doi.org/10.1364/oe.19.026627.

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32

Jiang, Yang, Guangfu Bai, Lin Hu, Chuang Ma, Yuanyuan Shan, Jianhui Liang, and Hongxia Li. "Subharmonic Generation in a Bandwidth Clamped Actively Mode-Locked Fiber Laser." IEEE Photonics Technology Letters 25, no. 23 (December 2013): 2331–34. http://dx.doi.org/10.1109/lpt.2013.2286172.

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33

Popov, V., E. Shandarov, and S. Shandarov. "Generation of spatial subharmonic gratings through mode mixing in planar waveguides." Journal of the Optical Society of America B 9, no. 9 (September 1, 1992): 1661. http://dx.doi.org/10.1364/josab.9.001661.

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34

Fang, Chung-Chieh. "Unified subharmonic oscillation conditions for peak or average current mode control." International Journal of Circuit Theory and Applications 43, no. 8 (March 20, 2014): 995–1014. http://dx.doi.org/10.1002/cta.1989.

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35

Nguyen, T. T., and W. Derek Humpage. "z-plane load modelling with particular reference to subharmonic mode responses." Electric Power Systems Research 16, no. 1 (January 1989): 33–45. http://dx.doi.org/10.1016/0378-7796(89)90035-7.

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36

Pelekasis, Nikolaos A., and John A. Tsamopoulos. "Bjerknes forces between two bubbles. Part 2. Response to an oscillatory pressure field." Journal of Fluid Mechanics 254 (September 1993): 501–27. http://dx.doi.org/10.1017/s002211209300223x.

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The motion of two gas bubbles in response to an oscillatory disturbance in the ambient pressure is studied. It is shown that the relative motion of bubbles of unequal size depends on the frequency of the disturbance. If this frequency is between the two natural frequencies for volume oscillations of the individual bubbles, the two bubbles are seen to move away from each other; otherwise attractive forces prevail. Bubbles of equal size can only attract each other, irrespective of the oscillation frequency. When the Bond number, Bo (based on the average acceleration) lies above a critical region, spherical-cap shapes appear with deformation confined on the side of the bubbles facing away from the direction of acceleration. For Bo below the critical region shape oscillations spanning the entire bubble surface take place, as a result of subharmonic resonance. The presence of the oscillatory acoustic field adds one more frequency to the system and increases the possibilities for resonance. However, only subharmonic resonance is observed because it occurs on a faster timescale, O(1/ε), where ε is the disturbance amplitude. Furthermore, among the different possible periodic variations of the volume of each bubble, the one with the smaller period determines which Legendre mode will be excited through subharmonic resonance. Spherical-cap shapes also occur on a timescale O(1/ε). When the bubbles are driven below resonance and for quite large amplitudes of the acoustic pressure, ε ≈ 0.8, a subharmonic signal at half the natural frequency of volume oscillations is obtained. This signal is primarily associated with the zeroth mode and corresponds to volume expansion followed by rapid collapse of the bubbles, a behaviour well documented in acoustic cavitation experiments.

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37

Ji, C., N. Chubun, R. G. Broeke, J. Cao, Y. Du, P. Bjeletich, and S. J. B. Yoo. "Electrical subharmonic hybrid mode locking of a colliding pulse mode-locked laser at 28 GHz." IEEE Photonics Technology Letters 17, no. 7 (July 2005): 1381–83. http://dx.doi.org/10.1109/lpt.2005.848555.

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38

VUKASINOVIC, BOJAN, MARC K. SMITH, and ARI GLEZER. "Dynamics of a sessile drop in forced vibration." Journal of Fluid Mechanics 587 (August 31, 2007): 395–423. http://dx.doi.org/10.1017/s0022112007007379.

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The interfacial dynamics of a sessile water drop was investigated experimentally. The low-viscosity drop was forced by an underlying diaphragm driven vertically by a piezoelectric actuator. This high-frequency forcing produced very low diaphragm displacements, even at high acceleration amplitudes. As the driving amplitude was increased from zero, the drop exhibited several transitions to states of increasing spatio-temporal complexity. The first state of the forced drop consisted of harmonic axisymmetric standing waves that were present for even the smallest diaphragm motion. Wave modes up to 14 were observed an compared to theoretical results. As the forcing amplitude increased above a critical value, a parametrically driven instability occurred that resulted in the appearance of subharmonic azimuthal waves along the contact line. The critical accelerations and the resulting wavenumbers of the azimuthal waves were documented. For larger values of the forcing amplitude, the subharmonic azimuthal waves coupled with the harmonic axisymmetric waves to produce a striking new lattice-like wave pattern. With a further increase in the forcing amplitude, the lattice mode disappeared and the interface evolved into a highly disordered state, dominated by subharmonic wave motion. The characteristics of the lattice and pre-ejection modes were documented with phase-locked measurements and spectral analysis. Finally, as the forcing amplitude increased above another critical value, the interface broke up via droplet ejection from individual wave crests.

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39

Chen, Shih-Chung, Chao-Lin Kuo, Chia-Hung Lin, Chih-Hung Hsu, and Chien-Kuo Tsui. "Applications of Fuzzy Sliding Mode Control for a Gyroscope System." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/931285.

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The study proposed the application of the fuzzy sliding mode for a gyroscope system status control. The state response analysis of the gyroscope system revealed highly nonlinear and chaotic subharmonic motions of2Tduring state formation. The current study discussed the use of tracking control on the sliding mode control and fuzzy sliding mode control of a gyroscope control system. Consequently, the gyroscope system drives from chaotic motion to periodic motion. The numerical simulation results confirm that the proposed controller provides good system stability and convergence without chattering phenomena.

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40

Sivasubramanian, Jayahar, and Hermann F. Fasel. "Numerical investigation of the development of three-dimensional wavepackets in a sharp cone boundary layer at Mach 6." Journal of Fluid Mechanics 756 (September 3, 2014): 600–649. http://dx.doi.org/10.1017/jfm.2014.434.

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AbstractDirect numerical simulations were performed to investigate wavepackets in a sharp cone boundary layer at Mach 6. In order to understand the natural transition process in hypersonic cone boundary layers, the flow was forced by a short-duration (localized) pulse. The pulse disturbance developed into a three-dimensional wavepacket, which consisted of a wide range of disturbance frequencies and wavenumbers. First, the linear development of the wavepacket was studied by forcing the flow with a low-amplitude pulse (0.001 % of the free-stream velocity). The dominant waves within the resulting wavepacket were identified as the second-mode axisymmetric disturbance waves. In addition, weaker first-mode oblique disturbance waves were also observed on the lateral sides of the wavepacket. In order to investigate the nonlinear transition regime, large-amplitude pulse disturbances (0.5 % of the free-stream velocity) were introduced. The response of the flow to the large-amplitude pulse disturbances indicated the presence of a fundamental resonance mechanism. Lower secondary peaks in the disturbance wave spectrum were identified at approximately half the frequency of the high-amplitude frequency band, suggesting the possibility of a subharmonic resonance mechanism. However, the spectrum also indicated that the fundamental resonance was much stronger than the subharmonic resonance. A secondary stability investigation using controlled disturbances confirmed that fundamental resonance is indeed a dominant mechanism compared to subharmonic resonance. Furthermore, strong peaks in the disturbance wave spectrum were also observed for low-azimuthal-wavenumber second-mode oblique waves, hinting at a possible oblique breakdown mechanism. Thus, the wavepacket simulations indicate that the second-mode fundamental resonance and oblique breakdown mechanisms are the strongest for the investigated flow. Hence, both mechanisms are likely to be relevant in the natural transition process for a cone boundary layer at Mach 6.

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41

Chang, Chau-Lyan, and Mujeeb R. Malik. "Oblique-mode breakdown and secondary instability in supersonic boundary layers." Journal of Fluid Mechanics 273 (August 25, 1994): 323–60. http://dx.doi.org/10.1017/s0022112094001965.

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Laminar–turbulent transition mechanisms for a supersonic boundary layer are examined by numerically solving the governing partial differential equations. It is shown that the dominant mechanism for transition at low supersonic Mach numbers is associated with the breakdown of oblique first-mode waves. The first stage in this breakdown process involves nonlinear interaction of a pair of oblique waves with equal but opposite angles resulting in the evolution of a streamwise vortex. This stage can be described by a wave–vortex triad consisting of the oblique waves and a streamwise vortex whereby the oblique waves grow linearly while nonlinear forcing results in the rapid growth of the vortex mode. In the second stage, the mutual and self-interaction of the streamwise vortex and the oblique modes results in the rapid growth of other harmonic waves and transition soon follows. Our calculations are carried all the way into the transition region which is characterized by rapid spectrum broadening, localized (unsteady) flow separation and the emergence of small-scale streamwise structures. The r.m.s. amplitude of the streamwise velocity component is found to be on the order of 4–5 % at the transition onset location marked by the rise in mean wall shear. When the boundary-layer flow is initially forced with multiple (frequency) oblique modes, transition occurs earlier than for a single (frequency) pair of oblique modes. Depending upon the disturbance frequencies, the oblique mode breakdown can occur for very low initial disturbance amplitudes (on the order of 0.001% or even lower) near the lower branch. In contrast, the subharmonic secondary instability mechanism for a two-dimensional primary disturbance requires an initial amplitude on the order of about 0.5% for the primary wave. An in-depth discussion of the oblique-mode breakdown as well as the secondary instability mechanism (both subharmonic and fundamental) is given for a Mach 1.6 flat-plate boundary layer.

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42

Arahira, S., and Y. Ogawa. "480-GHz subharmonic synchronous mode locking in a short-cavity colliding-pulse mode-locked laser diode." IEEE Photonics Technology Letters 14, no. 4 (April 2002): 537–39. http://dx.doi.org/10.1109/68.992603.

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43

Kapat, Santanu. "Selectively Sampled Subharmonic-Free Digital Current Mode Control Using Direct Duty Control." IEEE Transactions on Circuits and Systems II: Express Briefs 62, no. 3 (March 2015): 311–15. http://dx.doi.org/10.1109/tcsii.2014.2369091.

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44

Hayashi, Rintaro, John S. Allen, Parag V. Chitnis, Jonathan Mamou, and Jeffrey A. Ketterling. "Subharmonic Response of Polymer Contrast Agents Based on the Empirical Mode Decomposition." IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 63, no. 12 (December 2016): 2107–13. http://dx.doi.org/10.1109/tuffc.2016.2615047.

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45

Dave, Jaydev K., Flemming Forsberg, Savitha Fernandes, Catherine W. Piccoli, Traci B. Fox, Daniel A. Merton, Lauren M. Leodore, and Anne L. Hall. "Static and Dynamic Cumulative Maximum Intensity Display Mode for Subharmonic Breast Imaging." Journal of Ultrasound in Medicine 29, no. 8 (August 2010): 1177–85. http://dx.doi.org/10.7863/jum.2010.29.8.1177.

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46

Kirrou, Ilham, and Mohamed Belhaq. "Effect of contact stiffness modulation in contact-mode AFM under subharmonic excitation." Communications in Nonlinear Science and Numerical Simulation 18, no. 10 (October 2013): 2916–25. http://dx.doi.org/10.1016/j.cnsns.2013.02.008.

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47

Popovic, Pavol, Ali H. Nayfeh, Kyoyul Oh, and Samir A. Nayfeh. "An Experimental Investigation of Energy Transfer from a High- Frequency Mode to a Low-Frequency Mode in a Flexible Structure." Journal of Vibration and Control 1, no. 1 (January 1995): 115–28. http://dx.doi.org/10.1177/107754639500100108.

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The objective of the present article is to experimentally observe and characterize the transfer of energy from low-amplitude, high-frequency modes to high-amplitude, low-frequency modes. The subject of the study is a three-beam frame. The excitation amplitude is restricted to below 2 g peak. The authors have focused on observing, characterizing, and documenting the excitation of the first mode by high-frequency forcing. The energy-transfer processes are identified by power spectra and characterized further by frequency and amplitude sweeps. The energy-transfer routes observed in the experiment are subharmonic resonance of order one-half, combination resonance of the additive type, and interaction between widely spaced modes. In the latter route, an excitation at a frequency that is more than 100 times the first-mode frequency has been observed to excite the first mode.

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48

Serra-Garcia, Marc, Miguel Molerón, and Chiara Daraio. "Tunable, synchronized frequency down-conversion in magnetic lattices with defects." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2127 (July 23, 2018): 20170137. http://dx.doi.org/10.1098/rsta.2017.0137.

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We study frequency conversion in nonlinear mechanical lattices, focusing on a chain of magnets as a model system. We show that, by inserting mass defects at suitable locations, we can introduce localized vibrational modes that nonlinearly couple to extended lattice modes. The nonlinear interaction introduces an energy transfer from the high-frequency localized modes to a low-frequency extended mode. This system is capable of autonomously converting energy between highly tunable input and output frequencies, which need not be related by integer harmonic or subharmonic ratios. It is also capable of obtaining energy from multiple sources at different frequencies with a tunable output phase, due to the defect synchronization provided by the extended mode. Our lattice is a purely mechanical analogue of an opto-mechanical system, where the localized modes play the role of the electromagnetic field and the extended mode plays the role of the mechanical degree of freedom. This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

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49

Kaiser, Joachim, Ingo Fischer, and Wolfgang Elsässer. "Mode locking of lateral modes in broad-area semiconductor lasers by subharmonic optical pulse injection." Applied Physics Letters 88, no. 10 (March 6, 2006): 101110. http://dx.doi.org/10.1063/1.2185252.

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50

Jones, C. A. "The transition to wavy Taylor vortices." Journal of Fluid Mechanics 157 (August 1985): 135–62. http://dx.doi.org/10.1017/s0022112085002336.

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The transition from steady axisymmetric Taylor vortices to time-dependent wavy vortices is examined. The critical Taylor number and frequency at the transition point are determined in the infinite-cylinder approximation for a wide range of parameters. The results are compared with long-aspect-ratio experiments. The variation with axial wavelength is examined, and is found to be important when the radius ratio η < 0.75. A new spatially subharmonic mode is found to be the most unstable mode in some parameter regimes. This mode is identified with the jet mode recently discovered experimentally by Lorenzen, Pfister & Mullin and by Cole.

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