Subharmonic Regime Research Articles

Nonlinear regimes of electric convection of poorly conducting fluids in weightlessness at low frequencies of an electric field

Electrothermal convection in a poorly conducting fluid in an alternating electric field of a horizontal capacitor is studied. The nonlinear regimes of electric convection in weightlessness are studies at low frequencies of the electric field on the basis of the pentamodal model within the framework of the electroconductive charge formation mechanism. Hysteresis transitions between two different synchronous and subharmonic regimes are found to exist. Transitions to chaos occur by means of either interemittance or a subharmonic cascade.

Different scenarios in sloshing flows near the critical filling depth

In the present paper, the sloshing flow in a cuboid tank forced to oscillate horizontally is investigated with both experimental and numerical approaches. The filling depth chosen is $h/L=0.35$ (with h the water depth and L the tank height), which is close to the critical depth. According to Tadjbakhsh & Keller (J. Fluid Mech., vol. 8, issue 3, 1960, pp. 442–451), as the depth passes through this critical value the response of the resonant sloshing dynamics changes from ‘hard spring’ to ‘soft spring’. The experimental tank has a thickness of $0.1L$ , reducing three-dimensional effects. High-resolution digital camera and capacitance wave probes are used for time recording of the surface elevation. By varying the oscillation period and the amplitude of the motion imposed on the tank, different scenarios are identified in terms of free-surface evolution. Periodic and quasi-periodic regimes are found in most of the frequencies analysed but, among these, sub-harmonic regimes are also identified. Chaotic energetic regimes are found with motions of greater amplitude. Typical tools of dynamical systems, such as Fourier spectra and phase maps, are used for the regime identification, while the Hilbert–Huang transform is used for further insight into doubling-frequency and tripling-period bifurcations. For the numerical investigation, an advanced and well-established smoothed particle hydrodynamics method is used to aid the understanding of the physical phenomena involved and to extend the range of frequencies investigated experimentally.

Symmetry properties and bifurcations of viscoelastic thermovibrational convection in a square cavity.

To fill a gap in the literature about the specific dynamics of thermovibrational flow in a square cavity filled with a viscoelastic fluid when vibrations and the imposed temperature gradient are concurrent, a parametric investigation has been conducted to investigate the response of this system over a relatively wide subregion of the space of parameters (Pr_{g}=10; viscosity ratio ξ=0.5; nondimensional frequency Ω=25, 50, 75, and 100; and Ra_{ω}∈[Ra_{ω,cr},3.3×10^{4}], where Ra_{ω,cr} is the critical vibrational Rayleigh number). Through solution of the governing nonlinear equationsformulated in the framework of the finitely extensible nonlinear elastic Chilcott-Rallison paradigm, it is shown that the flow is prone to develop a unique hierarchy of bifurcations where initially subharmonic spatiotemporal regimes can be taken over by more complex states driven by the competition of disturbances with different symmetries if certain conditions are considered. What drives a wedge between the cases with parallel and perpendicular vibrations is essentially the existence of a threshold to be exceeded to produce convection in the former case. Nevertheless, these two configurations share some interesting properties, which are reminiscent of the resonances and antiresonances typical of multicomponent mechanical structures. Additional insights into these behaviors are gained through consideration of quantities representative of the kinetic and elastic energy globally possessed by the system and its sensitivity to the initial conditions.

Harnessing adaptive bistable stiffness of hair-cell-bundle structure for broadband vibration applications

This study presents an initial study on the adaptive bistable stiffness of the hair cell bundle structure in a frog cochlea, and aims to harness its bistable nonlinearity that features a negative stiffness region for broadband vibration applications such as vibration-based energy harvesters. To this end, the mathematical model for describing the bistable stiffness is first formulated based on the modeling concept of piecewise type nonlinearities. The harmonic balance method was then employed to examine the nonlinear responses of bistable oscillator, mimicking hair cells bundle structure under the frequency sweeping condition, and their dynamic behaviors induced by bistable stiffness characteristics are projected on phase diagrams, and Poincare maps concerning the bifurcation. In particular, the bifurcation mapping at the super- and sub-harmonic regimes provides a better perspective to examine the nonlinear motions which occur in the biomimetic system. The use of bistable stiffness characteristics of hair cell bundle structure in frog cochlea thus offers physical insights to harness the adaptive bistable stiffness for metamaterial-like potential engineering structures such as vibration-based energy harvester, and isolator etc.

Structurally Stable Subharmonic Regime of a Driven Quantum Josephson Circuit

Driven quantum nonlinear oscillators, while essential for quantum technologies, are generally prone to complex chaotic dynamics that fall beyond the reach of perturbative analysis. By focusing on subharmonic bifurcations of a harmonically driven oscillator, we provide a recipe for the choice of the oscillator's parameters that ensures a regular dynamical behavior independently of the driving strength. We show that this suppression of chaotic phenomena is compatible with a strong quantum nonlinear effect reflected by the confinement rate in the degenerate manifold spanned by stable subharmonic orbits.

Sub-Harmonic Response Analysis of Nonlinear Dynamic Behaviors Induced by Piecewise-Type Nonlinearities in a Torsional Vibratory System

Piecewise-type nonlinearities, such as clutch dampers in a torsional system, induce complex nonlinear dynamic behaviors that resemble super- and sub-harmonic responses. This study focuses on investigating the sub-harmonic responses induced by piecewise-type nonlinearities in the middle of various dynamic behaviors in a torsional vibratory system. To examine the dynamic characteristics in a sub-harmonic regime, the harmonic balance method (HBM) was implemented. Its results were compared with the numerical simulation (NS). To reveal the sub-harmonic responses, the input conditions of the HBM were modified with a small number of input values. In addition, bifurcation diagrams were numerically determined and projected onto stable and unstable solutions of the HBM to examine the effective dynamic behaviors within the unstable regimes. The results of the HBM with the modified input conditions reveal the sub-harmonic effects well, and the comparisons of bifurcation diagrams under unstable conditions lead to an understanding of the complex dynamic behaviors. Overall, this study suggests the first analytical technique to determine the sub-harmonic responses with the HBM, and second investigates the complex dynamic behaviors in a practical vibratory system by considering the bifurcations in the unstable regimes.

Regimes identification of the viscous flow past an elliptic cylinder for Reynolds number up to 10000

In the present paper, the study of different regimes arising from the incompressible planar viscous flow past an elliptical cylinder is presented. In order to highlight the effect of the different parameters on the onset of the regimes, two different aspect ratios, 0.10 and 0.40, are considered and the angles of attack span from 0∘ to 90∘, while the Reynolds number is gradually increased from 100 to 10000. The analyses are focused on the lift force acting on the ellipse in order to identify the regime, with a consideration on the vorticity field patterns of the wake fields. The different regimes are investigated and the chaotic behaviour is established through different tools such as the Fourier spectra, phase maps and Poincaré sections. The investigation of the periodic regimes revealed several possible conditions in terms of lift time signal: monochromatic, non-monochromatic and sub-harmonic regimes. In addition, a quasi-periodic time behaviour with an underlying irregular amplitude modulation was also found among the test matrices performed. Increasing the Reynolds number the periodic regimes are lost, giving the place to chaotic behaviour. The numerical solutions are obtained through a vortex particle method called Diffused Vortex Hydrodynamics (DVH). Long time simulations have been carried out in order to guarantee the correct identification of the attained regime.

Period doubling bifurcation analysis and isolated sub-harmonic resonances in an oscillator with asymmetric clearances

In this paper, a frequency-domain characterization of the period doubling bifurcation is proposed. This allows an efficient detection and localization of such points along frequency response curves computed through continuation and the harmonic balance method. A simple strategy for branch switching to sub-harmonic regimes is presented as well. Furthermore, these bifurcations are tracked in a two-dimensional parameter space, and extremum points with respect to the tracking parameter are characterized and linked to sub-harmonic isola formation. As a test case for these methods, a forced Duffing oscillator with asymmetric clearances is studied numerically. The results, which include the prediction of period doubling cascades and sub-harmonic isolas, are then compared to experimental results, yielding an excellent agreement.

Effects of gap ratio on flow-induced vibration of two rigidly coupled side-by-side cylinders

Vortex-induced vibration of two elastically mounted and rigidly coupled side-by-side circular cylinders in uniform flow is investigated using Direct Numerical Simulations (DNS). The two cylinders are allowed to vibrate in the cross-flow direction only. The aim of this study is to investigate the effect of gap ratio between the two cylinders on the vibration and wake flow. To achieve this, simulations are performed for gap ratios of 0.5, 1 and 3 and a wide range of reduced velocities from 1 to 20. The Reynolds number, the mass ratio and the damping ratio are 1000, 2 and 0, respectively. For small gap ratios of 0.5 and 1, the range of reduced velocity with high vibration amplitude is divided into a lock-in regime and a subharmonic regime. The subharmonic regime is a regime where the vibration frequency is half the dominant frequency of each individual cylinder’s lift coefficient. In the subharmonic regime, the dominant frequency components of the two cylinders do not contribute to the vibration because they are out of phase with each other. In the lock-in and subharmonic regimes, the wake is vortex shedding flow instead of biased gap flow for small gap ratios of G=0.5 and 1. The combination of lock-in and subharmonic regimes at G=0.5 and 1 are wider than the lock-in regime of a single cylinder. The weak interference between the two cylinders make the lock-in regime of G=3 the same as that of a single cylinder. A single vortex street wider than that of a single cylinder is found for G=0.5 and two vortex streets are found for G=3. For G=1, vortex shedding happens but clearly defined vortex street does not exist because strong interaction between the vortices causes very fast dissipation of the vortices.

Breakup of Closed Curve - Quasiperiodic Route to Chaos in Vibroimpact System

At present chaotic vibrations are the one of the most interesting and explored subjects in nonlinear dynamics. Particularly the routes to chaos in non-smooth dynamical systems are of the special scientists’ interest. Quasiperiodic route to chaos in nonlinear non-smooth discontinuous 2-DOF vibroimpact system is studied in this paper. In narrowfrequency range different oscillatory regimes have succeeded each other many times under very small control parameter varying. There were periodic subharmonic regimes - chatters, quasiperiodic, and chaotic regimes. There were the zones of transition from one regime to another, the zones of prechaotic and postchaotic motion. The hysteresis effects (jump phenomena) occurred for increasing and decreasing frequencies. The chaoticity of obtained regime has been confirmed by typical views of Poincare map and Fourier spectrum, by the positive value of the largest Lyapunov exponent, and by the fractal structure of Poincare map.

Harmonic to subharmonic transition of the Faraday instability in miscible fluids

When a stable stratification between two miscible fluids is excited by a vertical and periodic forcing, a turbulent mixing zone can develop, triggered by the Faraday instability. The mixing zone grows and saturates to a recently predicted final value L sat [Grea and Ebo Adou, J. Fluid Mech. 837, 293 (2018)] when resonance conditions are no longer fulfilled. Notably, it is expected from the Mathieu stability diagram that the instability may evolve from a harmonic to a subharmonic regime for particular initial conditions. This transition is evidenced here in the full inhomogeneous system using direct numerical simulations with 1024 3 points: the analysis of one-point statistics and spectra reveals that turbulence is greatly enhanced after the transition, while the global anisotropy of both the velocity and concentration fields is significantly reduced. Furthermore, using the concept of sorted density field, we compute the background potential energy e b p of the flow, which increases only after the transition as a signature of irreversible mixing. While the gain in e b p strongly depends on the control parameters of the instability, the cumulative mixing efficiency is more robust. At saturation of the instability, available potential energy is partially released in the flow as background potential energy. Finally, it is shown numerically that for fixed parameters, a multiple-frequency forcing can modify the duration of the harmonic regime without significantly altering the asymptotic state.

Subharmonic anti-phase dynamics in coupled mode-locked semiconductor lasers.

We show that coupled mode-locked semiconductor lasers can operate in a subharmonic regime in which the two lasers pulsate in an anti-phase manner at one-half the fundamental mode-locking frequency of the solitary lasers. In the subharmonic mode, each pulse has almost twice the energy carried by the isolated lasers in the fundamental mode-locked regime and is also significantly shorter in duration. Depending on the unsaturated gain and coupling strength, the lasers can also exhibit bistability, perfect synchronization, and delayed synchronization, as well as three-halves and five-halves harmonic mode locking. The observed behaviors are robust and persist in the presence of noise.

Covert cavitation: Spectral peak suppression in the acoustic emissions from spatially configured nucleations.

Dual laser-nucleation is used to precisely configure two cavitation bubbles within a focused ultrasound field of f0 = 692 kHz, in proximity to the tip of a needle hydrophone. With both bubbles responding in the f0/2 sub-harmonic regime, confirmed via ultra-high speed shadowgraphic imaging, an emission spectrum with no sub-harmonic content is demonstrated, for an inter-bubble spacing ≈λ0. A spectral model for periodic shock waves from multiple nucleations demonstrates peak suppressions at nf0/2 when applied to the experiment, via a windowing effect in the frequency domain. Implications for single-element passive detection of cavitation are discussed.

Self‐compensation of DC–DC converters under peak current mode control

A new self-compensation technique is proposed for eliminating subharmonic oscillation and chaotic regimes in dc–dc switching converters under peak current mode control. The proposed method extracts a control signal from the error between the inductor current and a suitable reference and does not require an external signal generator as in the conventional ramp compensation scheme. The analytical expressions of the control domain using Filippov method are obtained. Simulation results show that the proposed technique can be implemented using standard analogue devices and can effectively eliminate subharmonic and chaotic oscillations and can ensure a stable operation for a wide range of the duty ratio.

Modeling, Dynamics, Bifurcation Behavior and Stability Analysis of a DC–DC Boost Converter in Photovoltaic Systems

A study of a DC–DC boost converter fed by a photovoltaic (PV) generator and supplying a constant voltage load is presented. The input port of the converter is controlled using fixed frequency pulse width modulation (PWM) based on the loss-free resistor (LFR) concept whose parameter is selected with the aim to force the PV generator to work at its maximum power point. Under this control strategy, it is shown that the system can exhibit complex nonlinear behaviors for certain ranges of parameter values. First, using the nonlinear models of the converter and the PV source, the dynamics of the system are explored in terms of some of its parameters such as the proportional gain of the controller and the output DC bus voltage. To present a comprehensive approach to the overall system behavior under parameter changes, a series of bifurcation diagrams are computed from the circuit-level switched model and from a simplified model both implemented in PSIM© software showing a remarkable agreement. These diagrams show that the first instability that takes place in the system period-1 orbit when a primary parameter is varied is a smooth period-doubling bifurcation and that the nonlinearity of the PV generator is irrelevant for predicting this phenomenon. Different bifurcation scenarios can take place for the resulting period-2 subharmonic regime depending on a secondary bifurcation parameter. The boundary between the desired period-1 orbit and subharmonic oscillation resulting from period-doubling in the parameter space is obtained by calculating the eigenvalues of the monodromy matrix of the simplified model. The results from this model have been validated with time-domain numerical simulation using the circuit-level switched model and also experimentally from a laboratory prototype. This study can help in selecting the parameter values of the circuit in order to delimit the region of period-1 operation of the converter which is of practical interest in PV systems.

Superharmonic resonances in a two-dimensional non-linear photonic-crystal nano-electro-mechanical oscillator

We investigate the non-linear mechanical dynamics of a nano-optomechanical mirror formed by a suspended membrane pierced by a photonic crystal. By applying to the mirror a periodic electrostatic force induced by interdigitated electrodes integrated below the membrane, we evidence superharmonic resonances of our nano-electro-mechanical system; the constant phase shift of the oscillator across the resonance tongues is observed on the onset of principal harmonic and subharmonic excitation regimes.

Numerical investigation of the nonlinear attenuation and dissipation of acoustic waves in a medium containing ultrasound contrast agents

The presence of microbubble ultrasound contrast agents (UCAs) in path of an ultrasound beam increases the attenuation of the medium. A detailed knowledge about the medium attenuation in the presence of UCAs is critical for optimizing ultrasound applications. In this study, a model incorporating the nonlinear attenuation of microbubbles is developed by deriving the complex and real part of the wave number from the Calfish model. Results showed that when UCAs are sonicated by their linear resonance frequency (fr), the effective attenuation of the medium can potentially decrease as the pressure increases, in agreement with experimental observations of the attenuation of monodisperse microbubbles. When sonicated with their pressure dependent resonance frequency (fs), the effective attenuation of the medium is smaller than the case of sonication with fr, but only below a pressure threshold (Ps). Above Ps, the effective attenuation increases abruptly (e.g. ~3 fold) and becomes significantly more than the attenuation during sonication with fr. The attenuation in the subharmonic (SH) regimes of oscillations (sonication with n*fr where n = 2, 3.) are considerably smaller compared to the cases of sonication with fr and fs (~10 times). The attenuation undergoes a sharp increase concomitant with the generation of SH oscillations.

Stability analysis of a high-step-Up DC grid-connected two-stage boost DC-DC converter

High conversion ratio switching converters are used whenever there is a need to step-up dc source voltage level to a much higher output dc voltage level such as in photovoltaic systems, telecommunications and in some medical applications. A simple solution for achieving this high conversion ratio is by cascading different stages of dc-dc boost converters. The individual converters in such a cascaded system are usually designed separately applying classical design criteria. However these criteria may not be applicable for the complete cascaded system . This paper first presents a glimpse on the bifurcation behavior that a cascade connection of two boost converters can exhibit. It is shown that the desired periodic orbit can undergo period doubling leading to subharmonic oscillations and chaotic regimes. Then, in order to simplify the analysis the second stage is considered as constant current sink and design-oriented analysis is carried out to obtain stability boundaries in the parameter space by taking into account slope interactions between the state variables in the two-different stages.

The influence of vibration on Marangoni waves in two-layer films

AbstractThe influence of time-periodic vibrations on long Marangoni waves in two-layer films is investigated. The problem is governed by a system of nonlinear equations obtained in the framework of the lubrication approximation. Periodic boundary conditions are applied on the boundaries of the computational region. The development of instabilities is investigated by means of nonlinear simulations. Excitation of two-dimensional and three-dimensional subharmonic wavy regimes is studied. A new phenomenon, the excitation of nonlinear waves with a temporal period that is four times larger than that of the gravity modulation, is revealed.

Vibrational dynamics of vocal folds using nonlinear normal modes

Many previous works involving physical models, excised and in vivo larynges have pointed out nonlinear vibration in vocal folds during voice production. Moreover, theoretical studies involving mechanical modeling of these folds have tried to gain a profound understanding of the observed nonlinear phenomena. In this context, the present work uses the nonlinear normal mode theory to investigate the nonlinear modal behavior of 16 subjects using a two-mass mechanical modeling of the vocal folds. The free response of the conservative system at different energy levels is considered to assess the impact of the structural nonlinearity of the vocal fold tissues.The results show very interesting and complex nonlinear phenomena including frequency–energy dependence, subharmonic regimes and, in some cases, modal interactions, entrainment and bifurcations.