Resonance Curves Research Articles

Stochastic Resonance and Bifurcation of Order Parameter in a Coupled System of Underdamped Duffing Oscillators

The long-term mean-field dynamics of coupled underdamped Duffing oscillators driven by an external periodic signal with Gaussian noise is investigated. A Boltzmann-type [Formula: see text]-theorem is proved for the associated nonlinear Fokker–Planck equation to ensure that the system can always be relaxed to one of the stationary states as time is long enough. Based on a general framework of the linear response theory, the linear dynamical susceptibility of the system order parameter is explicitly deduced. With the spectral amplification factor as a quantifying index, calculation by the method of moments discloses that both mono-peak and double-peak resonance might appear, and that noise can greatly signify the peak of the resonance curve of the coupled underdamped system as compared with a single-element bistable system. Then, with the input signals taken from laboratory experiments, further observations show that the mean-field coupled stochastic resonance system can amplify the periodic input signal. Also, it reveals that for some driving frequencies, the optimal stochastic resonance parameter and the critical bifurcation parameter have a close relationship. Moreover, it is found that the damping coefficient can also give rise to nontrivial nonmonotonic behaviors of the resonance curve, and the resultant resonant peak attains its maximal height if the noise intensity or the coupling strength takes the critical value. The new findings reveal the role of the order parameter in a coupled system of chaotic oscillators.

Interfacial Dynamics and Rheology of a Crude-Oil Droplet Oscillating in Water at a High Frequency.

We report investigations of a pendant diluted crude-oil droplet in water that is forced to oscillate at a frequency ω. The droplet interface contains a significant amount of surface-active agents and displays a marked viscoelastic rheology with elastic moduli larger than viscous ones. At a low frequency, fluid viscosity and inertia are negligible, which allows a direct determination of the dilatational interface rheology. At a large frequency, eigenmodes of inertial shape oscillations are excited. By decomposing the interface shape into spherical harmonics, the resonance curves of the inertial modes of the interface are determined, as well as the frequency and damping rate of each mode. These two parameters are of major importance for the prediction of the deformation and breakup of a droplet in any unsteady flow without any prior knowledge of either the chemical composition or the detailed rheological properties of the interface. Then, interfacial rheology is related to interface dynamics by solving the coupled dynamic equations for the two fluids and the interface. It turns out that the rheology of the interface is well described by an equivalent two-dimensional viscoelastic material, the elasticities and viscosities of which depend upon the frequency. A first significant result is that shear and dilatational elasticities are closely connected, as are shear and dilatational viscosities. This implies that intrinsic rheology plays a major role and that compositional rheology is either negligible or strongly coupled to the intrinsic one. A second major result is that, for moderately aged droplets (≤5000 s), the elasticity and viscosity at a high frequency (10-80 Hz) can be extrapolated from low-frequency measurements (≤1 Hz) by a simple power law of the frequency, ωz. The exponent z is related to the loss angle θloss by a relation found in many previous low-frequency investigations of crude-oil interfaces: z = θloss/2π. The present work thus extends classic observations obtained at a low frequency to a higher frequency range corresponding to the natural frequency of the droplets, where the droplet shape results from the balance between dynamic pressure and surface stresses and the interface involves simultaneous shear and dilatation. These results bring about serious constraints regarding the modeling of physicochemical underlying mechanisms and provide some insights for the understanding of the structure of crude-oil interfaces.

Nonlinear dynamics of self-, parametric, and externally excited oscillator with time delay: van der Pol versus Rayleigh models

The regular and chaotic vibrations of a nonlinear structure subjected to self-, parametric, and external excitations acting simultaneously are analysed in this study. Moreover, a time delay input is added to the model to control the system response. The frequency-locking phenomenon and transition to quasi-periodic oscillations via Hopf bifurcation of the second kind (Neimark–Sacker bifurcation) are determined analytically by the multiple time scales method up to the second-order perturbation. Approximate solutions of the quasi-periodic motion are determined by a second application of the multiple time scales method for the slow flow, and then, slow–slow motion is obtained. The similarities and differences between the van der Pol and Rayleigh models are demonstrated for regular, periodic, and quasi-periodic oscillations, as well as for chaotic oscillations. The control of the structural response, and modifications of the resonance curves and bifurcation points by the time delay signal are presented for selected cases.

Van der Pol-Duffing oscillator: Global view of metamorphoses of the amplitude profiles

Dynamics of the Duffing–van der Pol driven oscillator is investigated. Periodic steady-state solutions of the corresponding equation are computed within the Krylov-Bogoliubov-Mitropolsky approach to yield dependence of amplitude A on forcing frequency Ω as an implicit function, FA,Ω=0, referred to as resonance curve or amplitude profile.In singular points of the amplitude curve the conditions ∂F∂A=0, ∂F∂Ω=0 are fulfilled, i.e. in such points neither of the functions A=fΩ, Ω=gA, continuous with continuous first derivative, exists. Near such points metamorphoses of the dynamics can occur. In the present work the bifurcation set, i.e. the set in the parameter space, such that every point in this set corresponds to a singular point of the amplitude profile, is computed.Several examples of singular points and the corresponding metamorphoses of dynamics are presented.

Experimentally measuring an isolated branch of Nonlinear normal modes

Nonlinear normal modes (NNMs) are a key tool for investigating the behaviour of nonlinear dynamic systems. Previous work has shown that branches of NNMs can be isolated from other NNM responses, in a similar manner to isolated, or detached, resonance curves in the forced responses. Their isolated nature poses a significant challenge for the prediction and measurement of these NNM branches. This paper illustrates how isolated NNMs may exist in two-degree-of-freedom systems, with cubic nonlinearities, that exhibit a 1:3 resonance. This is first introduced using a general two-mass oscillator, before considering a two-mode reduced-order model of a continuous cross-beam structure that exhibits a coupling between its primary bending and torsional modes. In both cases, a combination of analytical and numerical techniques is used to show how the isolated NNM branch may evolve from a set of bifurcating NNM branches. A nonlinear force appropriation technique is used to experimentally measure the NNMs of the cross-beam structure. By comparing these measurements to the numerical studies, it is shown that some of these NNMs are on the isolated branch, representing the first experimental confirmation of isolated NNM branches.

Thickness-dependent magnetic anisotropy in obliquely deposited Fe(001)/Pd thin film bilayers probed by VNA-FMR**Project supported by the National Basic Research Program of China (Grant Nos. 2015CB921403 and 2016YFA0300701), the National Natural Science Foundation of China (Grant Nos. 51427801, 11374350, and 51671212), and the Chinese Government Scholarship (Grant No. 2015GXYG37).

The thickness-dependent magnetic anisotropy of obliquely deposited Fe(001)/Pd thin films on Mg(001) is investigated by fitting the field-dependent resonant field curve using the Kittel equation. In this study, three Fe film samples with thicknesses of 50 monolayers (ML), 45 ML, and 32 ML deposited at 0°, 45°, and 55°, respectively, are used. The magnetic anisotropy constant obtained from ferromagnetic resonance (FMR) spectra exhibits a dominant fourfold magnetocrystalline anisotropy (MCA) at the normal deposition angle with larger Fe thickness. However, the in-plane uniaxial magnetic anisotropy (UMA) is induced by a higher oblique deposition angle and a smaller thickness. Its hard axis lies between the [100] and [010] directions. The FMR data-fitting analysis yields a precise measurement of smaller contributions to the magnetic anisotropy, such as in-plane UMA. Due to MCA, when the magnetic field is weaker than the saturated field, the magnetization direction does not always align with the external field. The squared frequency-dependent resonant field measurement gives an isotropic Landé g-factor of 2.07. Our results are consistent with previous experiments conducted on the magneto-optical Kerr effect (MOKE) and anisotropic magnetoresistance (AMR) systems. Thus, a vector network analyzer ferromagnetic resonance (VNA-FMR) test-method for finding UMA in obliquely deposited Fe(001)/Pd bilayer ferromagnetic thin films, and determining the magnetic anisotropy constants with respect to the film normal deposition, is proposed.

The effect of the TiO2 film on the performance of the optical fiber SPR sensor

We studied the effect of the Titanium Oxide (TiO2) film on the performance of the optical fiber SPR with two kinds of probes. The light guiding properties and sensing performance of the D-type optical fiber SPR sensor coated with TiO2 was studied numerically by finite element method (FEM) and the sensing performance of the multimode-single mode-multimode (MSM) hetero-core optical fiber SPR sensor coated with TiO2 was investigated experimentally. The simulation and experimental results for two kinds of probes consistently show that the absence of TiO2 film layer on the gold film can enhance the refractive index sensitivity and tune the resonance curve of the sensor from visible light to near-infrared.

Broadside Coupled Split Ring Resonator as a Sensitive Tunable Sensor for Efficient Detection of Mechanical Vibrations

This paper explores the possibility of the precise determination of mechanical vibrations using metamaterial split ring resonator (SRR) structure. The amplitude of the interacting electromagnetic wave in the range of GHz frequency is directly varied in accordance with the amplitude of mechanical vibration, using a Broadside Coupled SRR (BCSRR) acting as a vibration sensor. Dependence of the spacing between the two rings on the resonance frequency of the BCSRR is used for the detection of vibration and it is achieved by allowing the spacing to change in accordance with the amplitude of mechanical vibration. For the effective sensing of mechanical vibration, the electromagnetic wave frequency is chosen at the center of the linear portion of the rising or the falling slope of the resonant curve of the BCSRR. By properly choosing the parameters of the BCSRR along with the effective tuning of the operating frequency, it is possible to detect even very weak vibrations. The chances of various distortions in the detected vibration waveform in connection with selection of the operating frequency and intensity of vibrations are also analyzed. The qualitative formulation of the detection process along with its experimental verification is presented. This novel method may find applications in the detection of mechanical vibrations caused by various man made and natural sources and may find manifold possibilities in the field of communication and instrumentation.

Mechanical model analysis for resonance shear measurement.

Resonance shear measurement (RSM), which we developed based on a surface force apparatus, can investigate the structuring, and rheological and tribological properties of confined liquids as a function of the surface separation distance (D) from several μm to zero with a nanometer resolution. Using RSM, we reported that the nanoconfined liquids, including commercial lubricants, exhibited properties quite different from the bulk phase. Mechanical modeling of the resonance system is necessary to quantitatively evaluate the properties of confined liquids and/or sheared interfaces. In this study, we improved the model for RSM as follows: (1) We directly measured the movements of the upper and lower surfaces in addition to the measurement on the movement of the vertical spring to confirm the model, which could be used to estimate the parameters used in the model; (2) we proposed a modified mechanical model which considers the effect of additional motion. This model could fit the resonance curves using a identical apparatus constant for the entire measurement range, while the characteristic values of the confined liquids are the same as those obtained using our previous model, and (3) we calculated the friction (shear) force using the improved model and obtained the characteristic values (viscous and elastic parameters) of the confined liquids. This study afforded the simplicity and reliability of a mechanical model analysis of resonance curves, and the friction force calculation employing the viscous and elastic terms can be useful for studying the mechanism of friction force.

Investigation of terahertz all-dielectric metamaterials.

The propagation properties of Si-based all-dielectric metamaterials (ADMs) structures were investigated systematically, taking into account the effects of structural parameters, operation frequencies, and graphene Fermi levels. The results manifested that ADMs indicated sharp resonant curves with large Q-factors of more than 60, and a figure of merit of approximately 20. Compared with that of thin metal metamaterial counterparts, the thickness of ADMs (in the range of tens of micrometers) required to excite obvious resonant curves was much larger. By introducing an asymmetrical structure, an obvious Fano-resonant peak was observed, which also became stronger with increasing asymmetrical degree. In addition, by unitizing a uniform graphene layer, the Fano-resonant curves can be flexibly modulated over a wide range, and the amplitude-modulation depth of the Fano peak was approximately 40% when the Fermi level varied in the range of 0.01-1.0 eV. These results are very useful for the design of high Q-factor dielectric devices in the future (e.g., biosensors, modulators, and filters).

Instability Classes of Interfacial Waves of Gravity-Capillary Infinite Depth

The instability of interfacial gravity-capillary waves is investigated. The main wave was generated by oblique reflection of a uniform wave train on a vertical wall. Fixing the density ratio at μ = 0.9, the depths: d1 = 0.1, d2 = 2, and the capillary at κ = 0.01, we have plotted the resonance curves of the four instability classes Ia, Ib, IIa, IIb , for all of the three configurations: Θ = 10: waves close to the stationary interfacial waves. Θ = 40: fully three-dimensional interfacial waves. Θ = 80: waves close to progressive interfacial waves.

Dual Polarized Multiband Stub and Slot Loaded Semicircular Microstrip Antenna

A multiband configuration using semicircular patch fabricated on glass epoxy at height of 1.58mm was reported. The patch was loaded with a stub and a slot. Three bands were observed respectively at 3.5GHz with gain of 1.5dBi due to patch, 2.45 GHz with gain of 1.2dBi due to stub and 5.5GHz with gain of 1.8dBi due to slot. However detailed explanation of various resonant modes for getting multiband dual polarized response was not given in the reported work. In this paper a detailed analysis using resonance curve plots, return loss plot and vector surface current distributions explaining the multiband dual polarized response of reported work is presented. It is observed that length variations of stub and slot modify various modes of semicircular patch in such a way that results into multiband band dual polarized response. The dimension along with position of slot and stub with given feed location causes matching input impedances of TM10, TM31 and TM32 modes which results in multiband dual polarized response.

Electric field tuning non-volatile dynamic magnetism in half-metallic alloys Co2FeAl/Pb(Mg1/3Nb2/3)O3-PbTiO3 heterostructure

The half-metallic characteristic of Co2FeAl Huesler alloy films in Co2FeAl/Pb(Mg1/3Nb2/3)O3-PbTiO3 heterostructure was controlled by different substrate temperatures during deposition. The evolution of microstructure shows the B2 ordering degree of Co2FeAl thin film increase with the substrate temperature decreasing from 550 °C to 480 °C during deposition, which can be verified by anisotropy magnetoresistance measurement. Moreover, the result measured by rotating-angle ferromagnetic resonance demonstrates B2 ordering degree of Co2FeAl thin film controlled the change of symmetry of magnetic anisotropy. In addition, the loop-like curve of magnetic resonance field/linewidth versus electric field exhibits the non-volatile behavior, which can be attributed to the 109° ferroelectric domain switching of PMN-PT substrate. This result can provide opportunities for magnetization control in multiferroic devices.

Evaluation of Polarization Characteristics of the Fiber-Pigtailed Waveguide-Type Ring Resonator and Implications for Resonant Micro-Optic Gyroscopes

A group of misaligned angles is proposed to describe the polarization characteristics of the fiber-pigtailed waveguide-type ring resonator (WRR). With these angles, a polarization analysis model is then set up using the Jones matrix method. The output polarization extinction ratios (PERs) of the WRR are obtained through measuring resonant curves under different input PERs. Though multiple numerical groups of solutions for misaligned angles can be obtained by substituting these input–output PERs into the polarization analysis model, we are delighted to find that different groups of misaligned angles do not affect the polarization analysis of the WRR. Thus, the output PER can be pre-evaluated by selecting any group of misaligned angles. The PER improvements of the output light from the WRR are compared both theoretically and experimentally when polarizers are added at the input and output ports of the cavity, as well as employing an intracavity polarizer. The influences of misaligned angles are also discussed.

Evaluation of gap influence on the dynamic response behavior of pump-turbine runner

Purpose The gaps between runner and nearby structures play an important role in the dynamic response of runner, especially for pump-turbines. This paper aims to evaluate the gap influence on the added mass and dynamic stress of pump-turbine runner and provide an improved method to predict the resonance of runner. Design/methodology/approach Acoustic-structural coupling method was used to evaluate the added mass factors of a reduced scale pump-turbine with different axial and radial gap size between runner and nearby rigid walls. Improved one-way fluid-structural interaction (FSI) simulation was used to calculate the dynamic stress of the runner, which takes into account fluid added mass effect. The time-dependent hydraulic forces on the runner surfaces that were obtained from unsteady CFD simulation were transferred to the runner structure as a boundary condition, by using mesh-matching algorithm at the FSI surfaces. Findings The results show that the added mass factors increase as the gap size decreases. The axial gaps have greater influence on the added mass factors for the in-phase (IP) modes than the counter-phase (CP) and crown-dominant (CD) modes, while the CP and CD modes are very sensitive to the radial gaps. The largest added mass factor is observed in (2 + 4)ND-CP mode (resonance mode). The results reveal that the transient structural dynamic stress analysis, with the consideration of gaps and fluid added mass, can accurately predict the resonance phenomenon. Resonance curve of the pump-turbine has been obtained which agrees well with the test result. The gap fluid has great influence on the resonance condition, while for non-resonance operating points, the effect of gaps on the dynamic stress amplitude is quite small. Originality/value This paper provides an accurate method to analyze the dynamic response during runner design stage for safety assessment. The resonance curve prediction has more significance than previous methods which predict the resonance of runner by modal or harmonic analysis.

Nonlinear breathing motions and instabilities of a pressure-loaded spherical hyperelastic membrane

This work presents the mathematical modeling, based on a variational formulation, for the analysis of the nonlinear static and dynamic breathing motions of a hyperelastic spherical membrane subjected to internal pressure. The membrane is composed of an isotropic, homogeneous, incompressible, and hyperelastic material that is modeled using the Mooney–Rivlin constitutive law. The equilibrium equations are obtained using the fully nonlinear elasticity theory. First, the static nonlinear analysis is performed, and the principal stretches and stresses are obtained as a function of the internal pressure. Additionally, a parametric analysis is conducted to study the influence of the material constants on the nonlinear equilibrium path and potential energy. Depending on the value of the two material constants, a bistable behavior may be observed, having a profound influence on the nonlinear vibrations of the membrane. Then, the equations of motion of the pressure-loaded membrane subjected to increasing pressure and an additional harmonically varying internal pressure are obtained, considering a nonlinear damping force. A detailed parametric analysis clarifies the influence of the Mooney–Rivlin material parameters and static preload on the natural frequencies and particularly on the nonlinear frequency–amplitude relation. Precise results for the backbone curves are obtained by the shooting method, using the fully nonlinear equation of motion, and compared with several approximations obtained by the harmonic balance methods, highlighting the influence of higher order nonlinear terms on the nonlinear response of hyperelastic membranes. Finally, a parametric analysis of the harmonically excited preloaded membrane is conducted to reveal the importance of the two-well potential function, damping and constitutive law on the resonance curves, bifurcation diagrams, and basins of attraction.

Measurement of interfacial wave dynamics in orbitally shaken cylindrical containers using ultrasound pulse-echo techniques

We present a novel experiment on interfacial wave dynamics in orbitally shaken cylindrical vessels containing two and three fluid layers. The experiment was designed as a hydrodynamical model for liquid metal batteries. It is intended to shed new light into some aspects of the very similar rotational wave motion emerging due to the metal pad roll instability, as the viscous damping behavior or the contact line dynamics. Both issues can be important to better predict instability onsets for upcoming liquid metal batteries and lab-size experiments. Different options are presented to realize stable and measurable multi-layer stratifications. We introduce a new acoustic measurement procedure allowing to reconstruct wave amplitudes also in opaque liquids by tracking ultrasonic pulse echoes reflected at the interfaces. Measurements of resonance curves and phase shifts were conducted for varying interface positions. A strong influence of the top and bottom walls was observed, considerably reducing wave amplitudes and eigenfrequencies, when the interface is getting close. Finally, measured resonance curves were successfully compared with an existing forced wave theory that we extended to two-layer interfacial waves. The comparison stresses the importance to carefully control the boundary condition at the contact line.

A reduced multimodal thermoelastic model of a circular Mindlin plate

A nonlinear thermoelastic model of a circular plate is presented in the paper. The model, based on the Mindlin plate theory, is extended by taking into account nonlinear geometrical terms. Partial differential equations of plate’s dynamics are derived for a fully coupled thermal and mechanical fields. Then the model is reduced to a set of ordinary differential equations taking into account the first three natural modes and assuming a constant thermal field. The influence of elevated temperature on the resonance curves and the mode involvement due to nonlinear and thermal couplings is presented. The analysis shows that the increased temperature may lead to various bifurcation scenarios. The buckling phenomenon and post-buckling nonlinear regular and chaotic oscillations are studied.

Study of Peculiarities of the Microwave Absorption Spectrum of Nanocrystalline Thin Magnetic Films

Based on the micromagnetic model which takes into account the random distribution of the uniaxial magnetic anisotropy directions in crystallites of a nanocrystalline film, an effective method has been implemented for calculation of the magnetization dynamics in microwave fields. For a certain range of crystallite sizes, when the energy of the random magnetic anisotropy is comparable to the exchange energy, a significant change of the ferromagnetic resonance field, broadening of the resonance line, and the appearance of an asymmetry in the shape of the resonance curve were found. With an increase of the crystallite sizes, the resonance field first grows, then, it quickly decreases to its minimum, and then, it grows again to reach saturation. In this case, the steepness of the left slope of the broadening resonance curve first decreases faster than that of the right slope, leading to the symmetry breaking of the resonance curve shape, then, the curve becomes symmetrical again, and then, the steepness of the left slope becomes greater than that of the right slope.

Analysis of microwave-readable RFTES bolometer

We demonstrated and analyzed the smooth microwave-driven transition of a superconducting bridge made from Hf into its normal state at bath temperatures below the critical temperature of Hf (Tc ≈ 380 mK). The bridge is integrated on a silicon chip with both the 600-700 GHz double-slot antenna and the 1.5-GHz CPW quarter-wave resonator (Q-factor ∼104) made from 100-nm Nb film. The experimental bridge was sized 2.5 um by 2.5 um by 50 nm and tested at temperatures down to 50 mK. Similar to the technique of MKID, we measured the dependence of transmission S21 on microwave power at the bottom of the resonance curve. It was found that the microwave power absorbed in the bridge fits to the model of hot electron gas, P∼Te6-Tph6. The internal NEP down to ≈10−18 W/⎷Hz is estimated due to thermal noise at the optimum electron gas temperature, Te ≈ 320 mK. The NEP can be scaled down below ≈10−19 W/√Hz via reasonable reduction of the bridge volume. The new detector circuit is suitable for integration within a large imaging array exploiting the FDM-readout.