Subharmonic Response Research Articles

Ambient Pressure Sensitivity of the Subharmonic Response of Coated Microbubbles: Effects of Acoustic Excitation Parameters.

The sensitivity of the acoustic response of microbubbles, specifically a strong correlation between their subharmonic response and the ambient pressure, has motivated development of a non-invasive subharmonic-aided pressure estimation (SHAPE) method. However, this correlation has previously been found to vary depending on the microbubble type, the acoustic excitation and the hydrostatic pressure range. In this study, the ambient pressure sensitivity of microbubble response was investigated. The fundamental, subharmonic, second harmonic and ultraharmonic responses from an in-house lipid-coated microbubble were measured for excitations with peak negative pressures (PNPs) of 50-700 kPa and frequencies of 2, 3 and 4 MHz in the ambient overpressure range 0-25 kPa (0-187 mmHg) in an in vitro setup. The subharmonic response typically has three stages-occurrence, growth and saturation-with increasing excitation PNP. We find distinct decreasing and increasing variations of the subharmonic signal with overpressure that are closely related to the threshold of subharmonic generation in a lipid-shelled microbubble. Above the excitation threshold, that is, in the growth-saturation phase, subharmonic signals decreased linearly with slopes as high as -0.56 dB/kPa with ambient pressure increase; below the threshold excitation (at atmospheric pressure), increasing overpressure triggers subharmonic generation, indicating a lowering of subharmonic threshold, and therefore leads to an increase in subharmonic with overpressure, the maximum enhancement being ∼11 dB for 15 kPa overpressure at 2 MHz and 100 kPa PNP. This study indicates the possible development of novel and improved SHAPE methodologies.

Experimental investigation of the direct and subharmonic responses of a new design of centrifugal pendulum vibration absorber

Centrifugal pendulum vibration absorbers (CPVAs) are often used by the automotive industry to reduce vibrations of the drivetrain. These passive devices consist of several masses oscillating along a given path relative to a rotor. Recent CPVA systems make use of rocking pendulums, meaning that the pendulums rotate about their centre of mass during their motion along their path. In this work, measurements on a new CPVA architecture made of ball-type rocking pendulums are performed in classical and subharmonic operations. To the authors’ knowledge, they lead to the first comparisons between experimental and analytical results regarding the nonlinear detuning of the rotor’s antiresonance and the saturation of the rotor’s response. It is also the first time this saturation phenomenon is observed experimentally. The CPVA investigated is shown to have a high filtering efficiency but its pendulums are subjected to slipping, which limits the operating range of the system. Experimental and analytical investigations of the slipping are carried-out to estimate the limit of adherence of the system.

Subharmonic fidelity revival in a driven PXP model

The PXP model hosts a special set of nonergodic states, referred to as quantum many-body scars. One of the consequences of quantum scarring is the periodic revival of the wave function fidelity. It has been reported that quantum fidelity revival occurs in the PXP model for certain product states, and periodic driving of the chemical potential can enhance the magnitude of quantum revival and can even change the frequencies of revival showing the subharmonic response. Although the effect of the periodic driving in the PXP model has been studied in the limit of certain perturbative regimes, the general mechanism of such enhanced revival and frequency change has been barely studied. In this paper, we investigate how periodic driving in the PXP model can systematically control the fidelity revival. Particularly, focusing on a product state called the N\'eel state, we analyze the condition of driving to enhance the magnitude of revival or change the frequencies of revival. To clarify the reason for such control, we consider the similarities between the PXP model and the free spin-$1/2$ model in graph theoretical analysis and show that the quantum fidelity feature in the PXP model is well explained by the free spin-$1/2$ model. In addition, under a certain limit of the driving parameters, an analytic approach to explain the main features of the fidelity revival is also applied. Our results give insight into the scarring nature of the periodically driven PXP model and pave the way to understand the (sub)harmonic responses in these models and controls thereof.

Constrained Parameter-Splitting Multiple-Scales Method for the Primary/Sub-Harmonic Resonance of a Cantilever-Type Vibration Energy Harvester

In this paper, the approximate analytical solutions obtained by using the constrained parameter-splitting-multiple-scales (C-PSMS) method to the primary and [Formula: see text] sub-harmonic resonances responses of a cantilever-type energy harvester are presented. The C-PSMS method combines the multiple-scales (MS) method with the harmonic balance (HB) method. Different from the erroneous stability results obtained by using the Floquet theory and the classical HB method, accurate stability results are obtained by using the C-PSMS method. It is found that the correction to the erroneous solution when the HB method and Floquet theory are adopted in the stability analysis of the primary and [Formula: see text] sub-harmonic resonances of a largely deflected cantilever-type energy harvester is necessary. On the contrary, the C-PSMS method gives much improved results compared to those obtained by using Floquet theory and HB method when the numbers of terms in each response expression are the same. The frequency response curves of the primary resonance and the [Formula: see text] sub-harmonic resonance of the harvester obtained by the C-PSMS method are compared to those obtained by the HB method and verified by those obtained by the fourth-order Runge–Kutta method. Moreover, the basin of attraction based on the fourth-order Runge–Kutta method is presented to confirm the inaccurate stability results obtained by using the HB method and Floquet theory. The convergence examinations on the stability analysis carried out by the HB method and Floquet theory show that enough terms in the response assumption are needed to achieve relatively accurate stability results when studying the stability of the primary and sub-harmonic resonances of a cantilever by using the HB method and the Floquet theory. However, the low-order C-PSMS method is able to give an accurate frequency-amplitude response and accurate stability results of the primary and sub-harmonic resonances of a largely deflected cantilever-type energy harvester.

Stability of Periodic Orbits and Bifurcation Analysis of Ship Roll Oscillations in Regular Sea Waves

Response, stability, and bifurcation of roll oscillations of a biased ship under regular sea waves are investigated. The primary and subharmonic response branches are traced in the frequency domain employing the Incremental Harmonic Balance (IHB) method with a pseudo-arc-length continuation approach. The stability of periodic responses and bifurcation points are determined by monitoring the eigenvalues of the Floquet transition matrix. The primary and higher-order subharmonic responses experience a cascade of period-doubling bifurcations, eventually culminating in chaotic responses detected by numerical integration (NI) of the equation of motion. Bifurcation diagrams are obtained through the period-doubling route to chaos. Solutions are aided with phase portrait, Poincaré map, time history and Fourier spectrum for better clarity as and when required. Finally, the same ship model is investigated under variable excitation moments that may result from different wave heights in regular seas. The biased ship roll model exhibits primary and subharmonic responses, jump phenomena, coexistence of multiple responses, and chaotically modulated motion. The stable, periodic, and steady-state roll responses obtained by the IHB method are validated by the NI method. Results obtained by both methods are found to agree very well.

Discrete Time Crystal Enabled by Stark Many-Body Localization.

Discrete time crystals (DTCs) have recently attracted increasing attention, but most DTC models and their properties are only revealed after disorder average. In this Letter, we propose a simple disorder-free periodically driven model that exhibits nontrivial DTC order stabilized by Stark many-body localization (MBL). We demonstrate the existence of the DTC phase by analytical analysis from perturbation theory and convincing numerical evidence from observable dynamics. The new DTC model paves a new promising way for further experiments and deepens our understanding of DTCs. Since the DTC order does not require special quantum state preparation and the strong disorder average, it can be naturally realized on the noisy intermediate-scale quantum hardware with much fewer resources and repetitions. Moreover, in addition to the robust subharmonic response, there are other novel robust beating oscillations in the Stark-MBL DTC phase that are absent in random or quasiperiodic MBL DTCs.

The subharmonic resonance of lipid-encapsulated ultrasound contrast agents

A salient feature of ultrasound contrast agent echo is their subharmonic response, a property that can be exploited for diagnostic contrast imaging. Here, we aim to explore the subharmonic behavior, specifically the subharmonic resonance, of two identical lipid-encapsulated microbubbles in close proximity to each other (center-to-center h = 6–16 μm) using finite element modeling. We simulated the subharmonic resonance response for bubbles (R0 = 0.5–1.5 μm) driven with 10 or 20 cycle tone bursts from 6 to 18 MHz under a peak negative pressure from 40to 180 kPa over a range of initial phospholipid packing values (σ0 = 0–0.01 N/m). Our results demonstrate that for increasing pressure, the transmit frequency at which the peak in subharmonic response is observed shifts monotonically towards higher frequencies (1–10%) for bubbles close to their buckling point, while the opposite trend occurs for bubbles with less dense initial packing (1–15%). For intermediate values of initial packing, a two-stage response is observed whereby the transmit frequency first decreases and then increases (5–9%) with increasing pressure. These results have implications in subharmonic-based diagnostic imaging techniques.

Delayed subharmonic response of a phospholipid-shelled ultrasound contrast agent: Effect of temperature and protein concentration

The subharmonic response of ultrasound contrast agents (UCAs) is important for imaging and therapy. Therefore, factors influencing the temporal evolution of subharmonic signals should be elucidated. We evaluated the influence of temperature (25°C and 37°C) and protein concentration in the fluid (0.5% w/v at 25°C and 37°C) on the subharmonic response of Definity, a phospholipid-shelled UCA. Definity was diluted to 106 microbubbles/ml and sonicated at 2 MHz frequency, 50 cycles, and 470 kPa peak rarefactional pressure. The subharmonic response demonstrated striking qualitative and quantitative changes with time. At 25°C, a delayed subharmonic onset was observed. The subharmonic signal appeared after 28 min, peaking at 18dB above the baseline. At 37°C, the subharmonic signal appeared immediately but it peaked (5 dB increase) after 52 min. In the presence of dissolved protein, the subharmonic signal appeared at 16 min at 25°C and at 4min at 37°C. The signal peaked at 24 dB and 10 dB above the baseline at 25°C and 37°C, respectively. These observations underscore the need to standardize acoustic measurements and may have implications for subharmonic imaging, non-invasive pressure sensing, and cavitation detection.

Time evolution of ambient pressure sensitivity of microbubbles subharmonic response as a function of hydrostatic pressure and the filling gas

The ultrasound contrast microbubbles have shown a great ability to produce nonlinear oscillations in response to acoustic excitation which results in the generation of harmonics and subharmonics of the excitation. The subharmonic response of ½ order changes significantly with a change in hydrostatic pressure in the physiological range. This property is being investigated for subharmonic aided pressure estimation (SHAPE), a noninvasive alternative to current methods for vascular pressure measurement in critical organs. In this study, the ambient pressure sensitivity of the subharmonic response of microbubbles with different gas cores and shells was investigated over 25–700 kPa acoustic pressure and the hydrostatic pressure range of 0–25 kPa. The subharmonic showed different trends of increase and decrease with hydrostatic pressure. The sensitivity has shown gradual change over time which was observed to be related to the hydrostatic pressure magnitude and the gas core. These findings could help to calibrate the SHAPE.

Bichromatic visual stimulus with subharmonic response to achieve a high-accuracy SSVEP BCI system with low eye irritation

ObjectiveWhile various Steady State Visual Evoked Potential Brain-Computer Interfaces (SSVEP BCI) with very high performance have already been developed, eye irritation is a significant burden in their practical use. Generally, there is a tradeoff between eye irritation and BCI accuracy. However, this study proposes a bichromatic visual stimulus with low eye irritation while providing accurately detectable electroencephalogram responses. MethodsThe proposed visual stimulus, which is high-frequency in terms of luminance but with a subharmonic color component, was compared to the conventional high-frequency SSVEP on 16 subjects. Canonical Correlation Analysis, Least Absolute Shrinkage and Selection Operator, Multivariate Synchronization Index (MSI), and Extended MSI were used separately to recognize aimed visual stimuli in a 4-target BCI system. ResultsThe proposed stimulus resulted in significantly (p-value = 0.024) higher accuracy (96 ± 4.1%) compared to the reference (87.1 ± 14.6%) for a processing window of 4.5 s using the Extended MSI algorithm. Eye irritation scores (3.8 ± 1.0 on a 1–9 scale) were also significantly lower (p-value = 0.035) than that of the reference stimulus (4.3 ± 1.1). No cases of illiteracy were observed with the proposed waveform, whereas there were 2 with the reference stimulus. ConclusionThe proposed bichromatic waveform provided significantly higher accuracy, especially for users who do not have strong high-frequency SSVEP responses. Eye irritation was also significantly lower than with conventional reference visual stimulus. The proposed visual stimulus can generate electroencephalogram responses, not only in the high-frequency range but also at half the main frequency. SignificanceThe results are significant for realizing practical, accurate SSVEP BCI systems.

Floating hydroelastic circular plate in regular and irregular waves

An understanding of the hydroelastic response of a flexible circular plate to water waves is relevant to many problems in ocean engineering ranging from offshore wave energy converters and solar wind devices to very large floating structures such as floating airports and ice sheets. This paper describes results from physical model tests undertaken in the COAST laboratory at the University of Plymouth. Response amplitude operators (RAOs) of a floating flexible circular disk are determined for incident monochromatic and irregular wave trains, the latter defined by JONSWAP spectra. Free-surface displacements are measured using wave gauges, and the plate motion recorded using a QUALISYS® motion tracking system. Different basin depths and plate thicknesses are considered in order to quantify the effects of water depth and flexural plate rigidity on the overall dynamic behaviour of the circular disk. We present synchronous and subharmonic nonlinear responses for monochromatic waves, and displacement spectra for irregular waves. The measured wave hydrodynamics and disk hydroelastic responses match theoretical predictions based on linear potential flow theory.

Magneto-Aero-Elastic Superharmonic and Subharmonic Resonances, Bifurcations and Chaos of Conductive Spinning Circular Plates

This paper focuses on the magneto-aero-elastic nonlinear resonances, bifurcations and chaos for superharmonic and subharmonic responses of conductive circular plates spinning in air and magnetic fields. The electromagnetic forces acting on in-plane are presented by using electromagnetic field theory, and a rotational dissipative damping aerodynamic model is employed to characterize the aerodynamic forces acting on the plates. Considering the effects of the geometric nonlinearity and centrifugal tensioning, the magneto-aero-elastic governing equations of spinning conductive circular plates are derived by the Hamiltonian principle. Based on the mode shapes with Bessel functions, the transverse nonlinear vibration differential equations of spinning plates with clamped-edge are derived by the Galerkin method. Amplitude-frequency equations for subharmonic and superharmonic responses are obtained by using the approach of multiple scales, respectively. Numerical calculations indicate the influences of important parameters (e.g. magnetic field intensity, excitation amplitude and rotational speed) on amplitude frequency characteristics. Moreover, it is demonstrated by the numerical results that system responses are sensitive to the variation of these important parameters, where the complex intermittency, period-doubling and chaotic dynamical behaviors exist.

An Ising machine based on networks of subharmonic electrical resonators

Combinatorial optimization problems are difficult to solve with conventional algorithms. Here we explore networks of nonlinear electronic oscillators evolving dynamically towards the solution to such problems. We show that when driven into subharmonic response, such oscillator networks can minimize the Ising Hamiltonian on non-trivial antiferromagnetically-coupled 3-regular graphs. In this context, the spin-up and spin-down states of the Ising machine are represented by the oscillators’ response at the even or odd driving cycles. Our experimental setting of driven nonlinear oscillators coupled via a programmable switch matrix leads to a unique energy minimizer when one exists, and probes frustration where appropriate. Theoretical modeling of the electronic oscillators and their couplings allows us to accurately reproduce the qualitative features of the experimental results and extends the results to larger graphs. This suggests the promise of this setup as a prototypical one for exploring the capabilities of such an unconventional computing platform.

Subharmonic responses of a nonlinear energy harvester with an asymmetric double well potential

A nonlinear energy harvester with asymmetric potential wells is studied numerically. An energy transducing piezoelectric patch is attached to the cantilever elastic beam to harvest kinetic energy of the moving frame. To obtain an asymmetric effect in the double well potential induced by permanent magnets, an elastic ferromagnetic beam resonator misaligned with respect to the magnets' distribution axis of symmetry was used. The system response under the frequency-swept excitation reporting bifurcations to subharmonic and chaotic solutions was analysed. The sensitivity of the system solutions to the parameters responsible for potential asymmetry and electro-mechanical coupling was also investigated.

Finite strain-based theory for the superharmonic and subharmonic resonance of beams resting on a nonlinear viscoelastic foundation in thermal conditions, and subjected to a moving mass loading

We address the nonlinear free vibration, superharmonic and subharmonic resonance response of homogeneous Euler–Bernoulli beams resting on nonlinear viscoelastic foundations, under a moving mass and an abrupt uniform temperature rise. The nonlinear differential equation of motion stemming from the Hamiltonian principle and Finite Strain Theory is discretized according to a Galerkin decomposition method, and is solved by means of a multiple time scale method. A comparison between the Finite Strain theory and the Von-Karman approach is discussed, accounting for the effect of temperature rise, linear and nonlinear coefficients of the elastic foundation on the nonlinear vibration history and phase trajectory. At the same time, we check for the sensitivity of the frequency response of the system in superharmonic and subharmonic resonance for different input parameters, namely, location, velocity, and magnitude of the moving load, temperature rise and elastic foundation.

Subharmonic Resonance of Duffing Oscillator With Dry Friction Under Foundation Excitation

Abstract The 1/3 subharmonic resonance response of Duffing oscillator with Coulomb dry friction under foundation excitation is investigated, and the approximate analytical solution of the subharmonic resonance of the system is obtained by using the incremental averaging method. Based on the approximate analytical solution of the primary resonance obtained by the averaging method, the approximate analytical solution of subharmonic resonance is solved by the averaging method according to the incremental equation, and the amplitude–frequency response equation of subharmonic resonance is obtained. It is found that the Coulomb friction affects the amplitude–frequency response of both the primary resonance and subharmonic resonance of the nonlinear dry friction system in the form of equivalent damping. The comparison between the approximate analytical solution and the numerical solution shows that the approximate analytical solutions of the primary resonance and subharmonic resonance are both in very good agreement with the numerical solution. The existence condition of the 1/3 subharmonic resonance for the nonlinear dry friction system is presented, and the stability of the steady-state solution of subharmonic resonance is also judged. Based on the approximate analytical solution, the effects of the nonlinear stiffness and the Coulomb friction on the amplitude–frequency response of resonance and critical frequency of 1/3 subharmonic resonance of the nonlinear dry friction system are analyzed in detail. The analysis results show that the incremental averaging method can effectively obtain the approximate analytical solution in unified form for the subharmonic resonance of nonlinear system with Coulomb friction.

Digital quantum simulation of Floquet symmetry-protected topological phases

Quantum many-body systems away from equilibrium host a rich variety of exotic phenomena that are forbidden by equilibrium thermodynamics. A prominent example is that of discrete time crystals1–8, in which time-translational symmetry is spontaneously broken in periodically driven systems. Pioneering experiments have observed signatures of time crystalline phases with trapped ions9,10, solid-state spin systems11–15, ultracold atoms16,17 and superconducting qubits18–20. Here we report the observation of a distinct type of non-equilibrium state of matter, Floquet symmetry-protected topological phases, which are implemented through digital quantum simulation with an array of programmable superconducting qubits. We observe robust long-lived temporal correlations and subharmonic temporal response for the edge spins over up to 40 driving cycles using a circuit of depth exceeding 240 and acting on 26 qubits. We demonstrate that the subharmonic response is independent of the initial state, and experimentally map out a phase boundary between the Floquet symmetry-protected topological and thermal phases. Our results establish a versatile digital simulation approach to exploring exotic non-equilibrium phases of matter with current noisy intermediate-scale quantum processors21.

Examination of sub-harmonic responses along with various initial conditions induced by multi-staged clutch damper system

Using the harmonic balance method to investigate the nonlinear dynamic behaviors pertaining to sub-harmonic responses is difficult compared with that of super-harmonic cases because of the limitations of the HBM. Since sub-harmonic motions differ under various initial conditions, difficulties can arise when this method is used to calculate all possible solutions within sub-harmonic resonances. To explore complex dynamic behaviors in sub-harmonic resonant areas, this study suggests mathematical and numerical techniques to estimate sub-harmonic responses depending on various initial conditions. First, sub-harmonic responses are calculated under various excitation conditions relevant to the sub-harmonic input locations of the HBM formula. Second, the HBM results are verified by comparing them with the results of the numerical simulation (NS) under various initial conditions with respect to different frequency up-sweeping paths. Finally, the positive real part of the eigenvalues is examined to anticipate bifurcation characteristics, which reflect the relevance of the complex dynamic behaviors in the eigenvalues’ unstable solutions. Overall, this study successfully proves that the techniques and methods described are suitable for examining complex sub-harmonic responses, and suggests basic ideas for analyzing nonlinear dynamic behaviors in sub-harmonic resonances using the HBM.

Subharmonic centrifugal pendulum vibration absorbers allowing a rotational mobility

Rotating machines are often subjected to fluctuating torques, leading to vibrations of the rotor and finally to premature fatigue and noise pollution. This work addresses a new design of centrifugal pendulum vibration absorbers (CPVAs), used to reduce the vibrations in an automotive transmission line. These passive devices, composed of several masses oscillating along a trajectory relative to the rotor, are here tuned at a subharmonic of the targeted harmonic torque frequency. Thanks to the inherent non-linearities, a CPVA with two masses oscillating in phase opposition is able to efficiently counteract the input torque, with particular features such as saturation phenomena. This work particularly extends previous works to a new class of CPVA, whose peculiarity is that masses admit a significant rotation motion relative to the rotor, thus adding the benefit of their rotatory inertia. Results on the system’s subharmonic response and its stability are obtained thanks to an analytical perturbation method, and design guidelines are proposed. The validity of those results is also confirmed through comparisons with numerical solutions and the performance of this subharmonic system is compared to that of a classical CPVA tuned at the torque frequency.

Dynamics of Colombo’s Top: non-trivial oblique spin equilibria of super-Earths in multiplanetary systems

ABSTRACT Many Sun-like stars are observed to host close-in super-Earths (SEs) as part of a multiplanetary system. In such a system, the spin of the SE evolves due to spin–orbit resonances and tidal dissipation. In the absence of tides, the planet’s obliquity can evolve chaotically to large values. However, for close-in SEs, tidal dissipation is significant and suppresses the chaos, instead driving the spin into various steady states. We find that the attracting steady states of the SE’s spin are more numerous than previously thought, due to the discovery of a new class of ‘mixed-mode’ high-obliquity equilibria. These new equilibria arise due to subharmonic responses of the parametrically driven planetary spin, an unusual phenomenon that arises in non-linear systems. Many SEs should therefore have significant obliquities, with potentially large impacts on the physical conditions of their surfaces and atmospheres.