Parametric Resonance Research Articles

Generalization of the Faraday problem for the mechanical system «reservoir – liquid» in the presence of a vertical periodic (sawtooth) disturbance

The generalization of Faraday's classic problem about the parametric resonance of the free surface during vertical oscillations of the tank according to the periodic law of the modulus of the cosine is considered. Regions of stability and instability are designed, which are compared with similar results for the classical Faraday problem. The behavior of the system is considered on the basis of a nonlinear multimode model, which describes the combined movement of the tank and fluid under the influence of external force excitation or kinematic disturbance.

Parametric resonance brain model

The paper introduces a parametric resonance model for characterizing some features of the brain’s electrical activity. This activity is assumed to be a fundamental aspect of brain functionality underpinning functions from basic sensory processing to complex cognitive operations such as memory, reasoning, and emotion. A pivotal element of the proposed parametric model is neuron synchronization which is crucial for generating detectable brain waves. The analysis of the frequency content of brain waves, categorized as delta (0÷4 Hz), theta (4÷7 Hz), alpha (8÷12 Hz), beta (13÷30 Hz), and gamma (30÷100 Hz) reveals, notably, that the mean frequency of each of these brain wave classes is, in sequence, approximately the double of that of the previous one. Based on this observation, the proposed parametric resonance model suggests a cascade of amplification effects. Following the proposed model, in the transition from wakefulness to sleep, the brain wave bands are energized at double frequency by higher frequency neighboring bands; on the contrary, in the sleep to awake transition, brain waves are energized at a half frequency by their lower frequency neighbor waves. Finally, the trend of increasing amplitude values from higher to lower frequencies lends empirical support to the parametric resonant brain model validity.

The influence of the quasi-electrostatic support on the amplification of space charge waves in the amplification section of a superheterodyne free electron laser

Abstract In this work, a theoretical study of the influence of a quasi-electrostatic support on the amplification level of the slow space charge wave (SCW) in the amplification section of a superheterodyne free electron laser (FEL) was carried out. One of the ways to significantly increase the saturation level of the slow SCW is maintaining the conditions of a three-wave parametric resonance between the slow, fast SCWs and the resulting pump electric field. This can be done by introducing the quasi-electrostatic support in the superheterodyne FEL amplification section. Also, it was found that the generated pump electric field significantly influences the maintenance of parametric resonance conditions. As a result, this increases the saturation level of the slow SCW by 70%. Finally, the quasi-electrostatic support significantly reduces the maximum value of the electrostatic undulator pump field strength, which is necessary to achieve the maximum saturation level of the slow SCW.

Nonlinear deterministic dynamic analysis of a GPL micropipe conveying pulsating laminar flow

In several industries, the handling of fluid-filled micropipes is common. New-brand structures are increasingly being manufactured using advanced materials. This article tackles one of the crucial dynamic analyses that an advanced fluid-filled micropipe encounters. The dynamics of a graphene nanoplatelets-reinforced micropipe (GPL micropipe) under pulsating laminar flow for the first time is examined. The Euler-Bernoulli beam model follows the modified couple stress theory (MCST) and von-Karman’s nonlinear strain to formulate the problem. The impression of the flow profile exerted by a real flow as a consequence of fluid viscosity, is taken into account. The plug flow model results are compared to those regarding real flow consideration. Using the method of multiple scales (MMS), we determine the instability area and the steady state response of the GPL micropipe subjected to the principal parametric resonance of one of its modes due to pulsating laminar flow by applying this method to the discretized couple nonlinear gyroscopic governing equations. The Floquet theory treats the linear equations and fourth-order Runge–Kutta (RK4) method attacks nonlinear ones to validate MMS findings. It is confirmed that the 0.3 % addition of the GPL to matrix phase significantly enhances its advanced attributes, making it a highly effective material. The GPL reinforcement phase in the FGO pattern increases the critical flow velocity that exhibits the micropipe to static instability by 37.98 % , and critical flow stimulation amplitude fraction by 152.19 % , while declines instability area bandwidth by 36.95 % , and steady state response by 67.17 % relative to a non-reinforced micropipe. A denser flow degrades the corresponding values by 18.40 % , 42.59 % , 41.67 % , and 227.28 % in comparison to a lighter fluid. The declared findings provide crucial insights into the key design elements affecting significantly the functioning of fluid-filled GPL micropipes system, and regulate future research and expand openings for industry applications.

Inflaton production of scalar dark matter through fluctuations and scattering

We study the effects on particle production of a Planck-suppressed coupling between the inflaton and a scalar dark matter candidate, χ. In the absence of this coupling the dominant source for the relic density of χ is the long wavelength modes produced from the scalar field fluctuations during inflation. In this case, there are strong constraints on the mass of the scalar and the reheating temperature after inflation from the present-day relic density of χ (assuming χ is stable). When a coupling σϕ2χ2 is introduced, with σ=σ˜mϕ2/MP2∼10−10σ˜, where mϕ is the inflaton mass, the allowed parameter space begins to open up considerably even for σ˜ as small as ≳10−7. For σ˜≳916, particle production is dominated by the scattering of the inflaton condensate, either through single graviton exchange or the contact interaction between ϕ and χ. In this regime, the range of allowed masses and reheating temperatures is maximal. For 0.004<σ˜<50, constraints from isocurvature fluctuations are satisfied, and the production from parametric resonance can be neglected. Published by the American Physical Society 2024

Non-Bloch band theory for time-modulated discrete mechanical systems

This study establishes a non-Bloch band theory for time-modulated discrete mechanical systems. We consider simple mass–spring chains whose stiffness is periodically modulated in time. Using the temporal Floquet theory, the system is characterized by linear algebraic equations in terms of Fourier coefficients. This allows us to employ a standard linear eigenvalue analysis. Unlike non-modulated linear systems, the time modulation makes the coefficient matrix non-Hermitian, which gives rise to, for example, parametric resonance, non-reciprocal wave transmission, and non-Hermitian skin effects. In particular, we study finite-length chains consisting of spatially periodic mass–spring units and show that the standard Bloch band theory is not valid for estimating their eigenvalue distribution. To remedy this, we propose a non-Bloch band theory based on a generalized Brillouin zone. This novel approach, the combination of the temporal Floquet theory for time-modulated systems and generalized Brillouin zone, enables the prediction of eigenvalue distribution under open boundary conditions and also quantitative characterization of non-Hermitian skin modes. The proposed theory is verified by some numerical experiments.

Parametric resonance of gravitational waves in general scalar-tensor theories

Gravitational waves offer a potent mean to test the underlying theory of gravity. In general theories of gravity, such as scalar-tensor theories, one expects modifications in the friction term and the sound speed in the gravitational wave equation. In that case, rapid oscillations in such coefficients, e.g. due to an oscillating scalar field, may lead to narrow parametric resonances in the gravitational wave strain. We perform a general analysis of such possibility within DHOST theories. We use disformal transformations to find the theory space with larger resonances, within an effective field theory approach. We then apply our formalism to a non-minimally coupled ultra-light dark matter scalar field, assuming the presence of a primordial gravitational wave background, e.g., from inflation. We find that the resonant peaks in the spectral density may be detectable by forthcoming detectors such as LISA, Taiji, Einstein Telescope and Cosmic Explorer.

Magneto-Thermoelastic Principal Parameter Resonance of a Functionally Graded Cylindrical Shell with Axial Tension

The principal parameter resonance of a ferromagnetic functionally graded (FG) cylindrical shell under the action of axial time-varying tension in magnetic and temperature fields is investigated. The temperature dependence of physical parameters for functionally graded materials (FGMs) is considered. Meanwhile, the tension bending coupling effect is eliminated by introducing the physical neutral surface. The kinetic and strain energies are gained with the Kirchhoff–Love shell theory. Based on the nonlinear magnetization characteristics of ferromagnetic materials, the electromagnetic force acting on the shell is calculated. The nonlinear vibration equations are obtained through Hamilton’s principle. Afterward, the vibration equations are discretized and solved by Galerkin and multiscale methods, respectively. The stability criterion for steady-state motion is established utilizing Lyapunov stability theory. After example analysis, the effects of magnetic field intensity, temperature and power law index on the static deflection are elucidated. Subsequently, the impacts of these parameters, as well as axial tension, on the amplitude-frequency characteristics, resonance amplitude, and multiple solution regions are discussed explicitly. Results indicate that the stiffness can be enhanced due to the generation of static deflection. The amplitude decreases with increasing magnetic field intensity, temperature, and power law index. When the magnetic field intensity surpasses a threshold, the resonance phenomenon disappears.

Vibration analysis associated with the operation of printing units in offset printing machines: applications towards metamaterials

The paper analyzes the vibrational behavior of cylinders in the offset printing machine caused by a cylinder gap shock. Specifically, it assesses the stability of a system of two cylinders. The analysis of the proposed model is reduced to solving a set of Hill equations. The singularity of the obtained equations is the relationship between the natural frequencies of the system and modulation depth. Numerical simulations, along with the generalized Hill’s determinant method, were employed to determine the critical parameters of parametric resonance, thereby establishing the conditions necessary for the stability of periodic vibrations.

Equilibrium Parametric Amplification in Raman-Cavity Hybrids.

Parametric resonances and amplification have led to extraordinary photoinduced phenomena in pump-probe experiments. While these phenomena manifest themselves in out-of-equilibrium settings, here, we present the striking result of parametric amplification in equilibrium. We demonstrate that quantum and thermal fluctuations of a Raman-active mode amplifies light inside a cavity, at equilibrium, when the Raman mode frequency is twice the cavity mode frequency. This noise-driven amplification leads to the creation of an unusual parametric Raman polariton, intertwining the Raman mode with cavity squeezing fluctuations, with smoking gun signatures in Raman spectroscopy. In the resonant regime, we show the emergence of not only quantum light amplification but also localization and static shift of the Raman mode. Apart from the fundamental interest of equilibrium parametric amplification, our Letter suggests a resonant mechanism for controlling Raman modes and thus matter properties by cavity fluctuations. We conclude by outlining how to compute the Raman-cavity coupling, and suggest possible experimental realizations.

Variable angle tow-steered fibres based rotating composite beam with rotary oscillations – Parametric excitation/instability analysis

This study is concerned with the stability analysis for various vibrational motions of tow-steered variable angle fibres based composite rotating beam with time varying speed. The formulation combines a higher order theory introducing sine function along with finite element methodology. It satisfies the plane stress condition in width direction of beam. The displacement kinematics are generic in the sense it includes various possible vibrational motions like chord and flap wise motions, torsional and axial vibration behaviours. The equilibrium equations for the proposed problem evolved adopting virtual dynamic work are then transformed into the equations of Mathieu-Hill type considering time varying harmonic rotational speed. Using Bolotin’s procedure coupled with C1 continuity-based beam element, the parametric resonance zones along with boundary limits are numerically evaluated. The eigenvalue type of solution methodology applied for the detailed study is validated for accuracy and computational efficiency against the time domain dynamic response analysis. Based on in-depth analysis, dynamic instability characteristics in terms of resonance range and origin of instability of composite beam subjected to the chord and flap wise, torsional and axial motions are studied considering curvilinear fibre path angles, beam cross section, hub radius, thickness ratio and mean rotary speed.

Parametric instability analysis of rotors under anisotropic boundary conditions

Ensuring rotor stability is a major concern in engineering, as instabilities can lead to catastrophic failures. Existing literature shows that anisotropic boundary conditions significantly affect the parametric instability characteristics of rotors under periodical axial loads. However, there is little literature systematically analyzing the formation mechanism of parametric resonance under these boundary conditions or providing a detailed classification of the parametric instability regions. Therefore, this paper presents a comprehensive parametric instability analysis of a rotor subjected to periodic axial loads under anisotropic boundary conditions. A novel approach based on the multiple scales method is proposed to address anisotropy in the boundary conditions. Using this approach, the analytical boundaries of the parametric instability regions are derived, and a proof regarding the absence of certain parametric resonances is presented. These analytical solutions are validated by numerical results obtained from the discrete transition matrix method, which form the basis for systematically investigating the effects of anisotropy in direct or cross-coupling stiffness/damping coefficients on the rotor instability. The key scientific contributions of this work include: Deriving analytical instability boundaries, providing a more efficient alternative to purely numerical methods while maintaining high accuracy; Demonstrating the absence of parametric resonance of difference type under both isotropic or anisotropic boundary conditions; Discovering that anisotropy in stiffness coefficients can induce self-interaction within a given forward or backward whirl mode, as well as interaction between two forward or two backward whirl modes, leading to additional instability regions; Reducing anisotropy in direct damping coefficients may increase critical dynamic load coefficients, potentially enhancing rotor safety; If the cross-coupling stiffness coefficients exceed the threshold for triggering intrinsic instability, the rotor may become unstable in all operating conditions. All these findings offer insights into the stability management of rotors under various operating conditions and provide valuable guidance for designing and operating safer, more efficient rotor systems.

Measuring Transverse Relaxation with a Single-Beam 894 nm VCSEL for Cs-Xe NMR Gyroscope Miniaturization.

The spin-exchange-pumped nuclear magnetic resonance gyroscope (NMRG) is a pivotal tool in quantum navigation. The transverse relaxation of atoms critically impacts the NMRG's performance parameters and is essential for judging normal operation. Conventional methods for measuring transverse relaxation typically use dual beams, which involves complex optical path and frequency stabilization systems, thereby complicating miniaturization and integration. This paper proposes a method to construct a 133Cs parametric resonance magnetometer using a single-beam vertical-cavity surface-emitting laser (VCSEL) to measure the transverse relaxation of 129Xe and 131Xe. Based on this method, the volume of the gyroscope probe is significantly reduced to 50 cm3. Experimental results demonstrate that the constructed Cs-Xe NMRG can achieve a transverse relaxation time (T2) of 8.1 s under static conditions. Within the cell temperature range of 70 °C to 110 °C, T2 decreases with increasing temperature, while the signal amplitude inversely increases. The research lays the foundation for continuous measurement operations of miniaturized NMRGs.

Sufficient condition for reliable logic operations in an over damped bistable system driven by Gaussian white noise.

In this paper, we investigate an overdamped bistable system subject to Gaussian white noise and logical inputs. By solving and estimating the steady distribution of the corresponding Fokker-Planck equation and considering the two essential features of the reliable logic operations (RLOs)-initial value independence and sign invariance-we establish the sufficient condition for RLO occurrence and identify parametric resonance phenomena in the system. Our numerical simulations confirm the reliability and accuracy of the theoretical results. This work offers insights into enhancing the accuracy of stochastic systems, particularly in the realm of logical stochastic resonance, thereby contributing to advancements in understanding and controlling stochastic dynamical systems.

VIBRATIONS OF A VERTICAL BEAM ROTATING WITH VARIABLE ANGULAR VELOCITY

An Euler-Bernoulli beam in vertical position rotating about its symmetry axis along its length is considered. The angular velocity is assumed to have small fluctuations about a constant mean velocity. The partial differential equation of motion is derived first. The equation is cast into a non-dimensional form. The natural frequencies are calculated for the pinned-pinned case. Principle parametric resonances such that the fluctuation frequency being close to two times one of the natural frequencies are considered. By employment of the Method of Multiple Scales, an approximate perturbation solution is found. The frequency response diagrams are drawn and the bifurcation points for transition from the trivial solution to the non-trivial solution are calculated. The conditions for which such resonances occur are exploited in the numerical results.

Radio-Frequency Magnetometry Based on Parametric Resonances.

We propose and demonstrate a radio-frequency (rf) atomic magnetometer based on parametric resonances. Previously, most rf atomic magnetometers are based on magnetic resonances and their sensitivities are often limited by spin-exchange relaxation. Here, we introduce a novel scheme for an rf magnetometer where the rf magnetic field is measured by exciting the parametric resonances instead of magnetic resonances using parametric modulation fields. In this way, the spin-exchange relaxation is almost eliminated. Benefiting from the low spin relaxation rate, the parametric resonance scheme exhibits a narrower linewidth and stronger signal, which results in a higher sensitivity. With a 6×6×3 mm^{3} Rb atomic vapor cell, we developed an rf atomic magnetometer with a noise floor of 2 fT/Hz^{1/2}, which is about one order of magnitude higher than the sensitivity achieved in the magnetic-resonance-based scheme. The presented rf detection scheme holds promise in advancing rf atomic magnetometers and brings new insight into their various applications.

Vibration-induced wall–bubble interactions under zero-gravity conditions

This work is devoted to a theoretical and numerical study of the dynamics of a two-phase system vapour bubble in equilibrium with its liquid phase under translational vibrations in the absence of gravity. The bubble is initially located in the container centre. The liquid and vapour phases are considered as viscous and incompressible. Analysis focuses on the vibrational conditions used in experiments with the two-phase system SF $_6$ in the MIR space station and with the two-phase system para-Hydrogen (p-H $_2$ ) under magnetic compensation of Earth's gravity. These conditions correspond to small-amplitude high-frequency vibrations. Under vibrations, additionally to the forced oscillations, an average displacement of the bubble to the wall is observed due to an average vibrational attraction force related to the Bernoulli effect. Vibrational conditions for SF $_6$ correspond to much smaller average vibrational force (weak vibrations) than for p-H $_2$ (strong vibrations). For weak vibrations, the role of the initial vibration phase is crucial. The difference in the behaviour at different initial phases is explained using a simple mechanical model. For strong vibrations, the average displacement to the wall stops when the bubble reaches a quasi-equilibrium position where the resulting average force is zero. At large vibration velocity amplitudes this position is near the wall where the bubble performs only forced oscillations. At moderate vibration velocity amplitudes the bubble average displacement stops at a finite distance from the wall, then large-scale damped oscillations around this position accompanied by forced oscillations are observed. Bubble shape oscillations and the parametric resonance of forced oscillations are also studied.

Axions and primordial magnetogenesis: the role of initial axion inhomogeneities

The relic density of dark matter in the ΛCDM model restricts the parameter space for a cosmological axion field, constraining the axion decay constant, the initial amplitude of the axion field and the axion mass. It is shown via lattice simulations how the relic density of axion-like particles with masses close to the one of the QCD axion is affected by axion-gauge field interactions and by initial axion inhomogeneities. For pre-inflationary axions, once the Hubble parameter becomes smaller than the axion mass, the latter starts to oscillate, and part of its energy density is spent producing gauge fields via parametric resonance. If the gauge fields are dark photons and Standard Model photons, the energy density of dark photons becomes higher than the one of the axion, while the high conductivity of the primordial plasma damps the oscillations of the photon field. Such a scenario allows for the production of small-scale, primordial magnetic fields, and it is found that the relic density of axions with a low decay constant are within the bounds set by the ΛCDM model, while GUT-scale axions are far too abundant. It is also shown that initial inhomogeneities of the axion field can change substantially the gauge field production, boosting or suppressing (depending on the axion parameters and couplings) the magnetogenesis mechanism with respect to an homogeneous axion field. It is found that when the axion mass is far lighter than the QCD axion model and the initial axion field is inhomogeneous, weak but cosmologically relevant magnetic field seeds can be generated on scales of the order of 0.1 kpc.

Space-time symmetry and nonreciprocal parametric resonance in mechanical systems.

Linear mechanical systems with time-modulated parameters can harbor oscillations with amplitudes that grow or decay exponentially with time due to the phenomenon of parametric resonance. While the resonance properties of individual oscillators are well understood, those of systems of coupled oscillators remain challenging to characterize. Here we determine the parametric resonance conditions for time-modulated mechanical systems by exploiting the internal symmetries arising from the real-valued and symplectic nature of classical mechanics. We also determine how these conditions are further constrained when the system exhibits external symmetries. In particular, we analyze systems with space-time symmetry where the system remains invariant after a combination of discrete translation in both space and time. For such systems, we identify a combined space-time translation operator that provides more information about the dynamics of the system than the Floquet operator does and use it to derive conditions for one-way amplification of traveling waves. Our exact theoretical framework based on symmetries enables the design of exotic responses such as nonreciprocal transport and one-way amplification in dynamic mechanical metamaterials and is generalizable to all physical systems that obey space-time symmetry.

Nonlinear vibrations of variable speed rotating graphene platelets reinforced blades subjected to combined parametric and forced excitation

It is generally acknowledged that the disturbance of rotating speed will cause parametric resonance for blades, and the presence of rotor displacement also can make the blade vibrate transversely. However, the coupled resonance mechanism of blades under the combined action of rotating speed disturbance and rotor displacement is not yet clear. To answer this question, this article investigates the nonlinear coupled resonance behavior of rotating blades in two cases: typical tuned (the frequency ratio of parametric and forced excitations equals 2:1) and frequency ratio detuned. A nonlinear vibration equation for rotating blades is established based on Euler beam theory in conjunction with geometric nonlinearity. The mixing rule and modified Halpin-Tsai model are used to calculate the effective properties of GPLRMF materials. Subsequently, using the method of varying amplitudes (MVA), an approximate analytical solution of the coupled resonance response is derived, the Jacobian matrix is used to determine the stability of non-trivial solutions, and the accuracy of the analytical results is verified with the aid of numerical solutions. Finally, the steady-state response of typical tuned and detuned coupled resonance is analyzed, and the influence mechanism of factors such as excitation amplitude, excitation phase angle and other parameters on coupled resonance is studied in detail. The results indicate that the supercritical coupled resonance curve exhibits double resonance peaks and jumps. Meanwhile, the steady-state response of coupled resonance highly depends on the excitation phase angle.