Nonlinear Mode Research Articles

Numerical simulation of freak wave generation and evolution process based on SPH method

With the occurrence of maritime accidents all over the world, the study of extreme waves has been promoted. These extreme waves are often called freak waves in the scientific community. At present, a variety of mechanisms are proposed to explain freak waves in the marine environment, such as directional focusing, dispersive focusing, and nonlinear modulation instability. At first, a hydrodynamic model is established based on the Smoothed Particle Hydrodynamics (SPH) method in the C# environment, and the regular wave generated by the piston wave generator is numerically simulated. Secondly, the simulation results were analyzed and visualized by using Paraview software, compared with physical experimental datas to verify the reliability of this model. Then, the generation and evolution of freak waves were numerically simulated through the movement of piston wave generators on both sides of a 2D numerical wave flume, and the influence of frequency and amplitude was analyzed by using directional focusing. The results show that if the frequency is smaller and the amplitude is larger, it is more likely to generate freak waves and wave-breaking phenomena. This study lays a foundation for the research of freak waves in the future.

Multiple resonance in coupled Duffing oscillators and nonlinear normal modes.

The resonance of N linearly coupled damped Duffing oscillators with a constant frequency sinusoidal driving force acting on the first oscillator is studied analytically by calculating the fixed points of the corresponding dynamical system and numerically using a fourth-order multivariate Runge-Kutta method. For a chain with N oscillators, we establish a general recursion scheme in the form of a system of equationsthat relates the amplitudes of the oscillators and the driving frequency, capable of describing resonance curves. We consider in detail the case of an oscillator chain with N=2 for high values of the driving amplitude and stiffness, and find hysteretical unstable regions in the resonance curves. In this unstable driving frequency regime, analysis of the time series reveals the presence of nonlinear normal modes visible as beating quasiperiodic oscillations.

Nonlinear Wave Damping by Kelvin–Helmholtz Instability-induced Turbulence

Magnetohydrodynamic kink waves naturally form as a consequence of perturbations to a structured medium, for example, transverse oscillations of coronal loops. Linear theory has provided many insights into the evolution of linear oscillations, and results from these models are often applied to infer information about the solar corona from observed wave periods and damping times. However, simulations show that nonlinear kink waves can host the Kelvin–Helmholtz instability (KHI), which subsequently creates turbulence in the loop, dynamics that are beyond linear models. In this paper we investigate the evolution of KHI-induced turbulence on the surface of a flux tube where a nonlinear fundamental kink mode has been excited. We control our numerical experiment so that we induce the KHI without exciting resonant absorption. We find two stages in the KHI turbulence dynamics. In the first stage, we show that the classic model of a KHI turbulent layer growing at ∝t is applicable. We adapt this model to make accurate predictions of the damping of the oscillation and turbulent heating as a consequence of the KHI dynamics. In the second stage, the now dominant turbulent motions are undergoing decay. We find that the classic model of energy decay proportional to t −2 approximately holds and provides an accurate prediction of the heating in this phase. Our results show that we can develop simple models for the turbulent evolution of a nonlinear kink wave, but the damping profiles produced are distinct from those of linear theory that are commonly used to confront theory and observations.

Magnetohydrodynamic waves in the partially ionized solar plasma.

This paper provides a comprehensive review of recent theoretical investigations concerning magnetohydrodynamic (MHD) waves in partially ionized solar plasma. First, we examine the properties of linear MHD waves in a uniform partially ionized plasma and discuss the relevant effects arising from partial ionization. Subsequently, we delve into MHD wave studies in the more intricate settings of the lower solar atmosphere and solar prominences. These investigations involve topics such as MHD waves in magnetic flux tubes, wave excitation, linear and nonlinear mode coupling and wave heating. We outline new challenges that future research should tackle. This article is part of the theme issue 'Partially ionized plasma of the solar atmosphere: recent advances and future pathways'.

Tracking superharmonic resonances for nonlinear vibration of conservative and hysteretic single degree of freedom systems

Many modern engineering structures exhibit nonlinear vibration. Characterizing such vibrations efficiently is critical to optimizing designs for reliability and performance. For linear systems, steady-state vibration occurs only at the forcing frequencies. However, nonlinearities (e.g., contact, friction, large deformation, etc.) can result in nonlinear vibration behavior including superharmonics — responses at integer multiples of the forcing frequency. When the forcing frequency is near an integer fraction of the natural frequency, superharmonic resonance occurs, and the magnitude of the superharmonics can exceed that of the fundamental harmonic that is externally forced. Characterizing such superharmonic resonances is critical to improving engineering designs. The present work extends the concept of phase resonance nonlinear modes (PRNM) to be applicable to general nonlinearities, and is demonstrated for eight different nonlinear forces. The considered forces include stiffening, softening, contact, damping, and frictional nonlinearities that have not been previously considered with PRNM. The proposed variable phase resonance nonlinear modes (VPRNM) method can accurately track superharmonic resonances for hysteretic nonlinearities that exhibit amplitude dependent phase resonance conditions that cannot be captured by PRNM. The proposed method allows for characterization of superharmonic resonances without constructing a full frequency response curve at every force level with the harmonic balance method. Thus, the present method allows for analysis of potential failures due to large amplitudes near the superharmonic resonance with reduced computational cost. The consideration of single degree of freedom systems in the present paper provides insights into superharmonic resonances and a basis for understanding internal resonances for multiple degree of freedom systems.

Experimental nonlinear model of a set of connecting elements in view of nonlinear modal coupling

The development process of mechanical systems involves the evaluation of its modes of vibrations in the frequency range of interest. In general, a linear modal analysis is sufficient to determine whether the system can operate in dynamic conditions. However, in some cases the assembly is composed of many subsystems connected through nonlinear connections which make the response depend on the amplitude and frequency of the excitation. In those cases, Linear Normal Modes (LNMs) are not sufficient to fully describe the dynamics of the system and Nonlinear Normal Modes (NNMs) must be used. Using substructuring techniques it is possible to treat nonlinear joints as independent subsystems. However, a reliable nonlinear model is needed to use this approach. The experimental characterization is the only way to correctly estimate the nonlinear behavior of the connection. Thus, the aim of this work is to develop and validate the experimental characterization to build an experimental nonlinear modal model of a strongly nonlinear element that can be used to connect different linear subsystems and can be regarded as a localized source of nonlinearity. The NNMs identification is performed using the single-point single-harmonic phase resonance method, and the experimental nonlinear modal model of the NLCE is obtained by retaining only the first harmonic term of each NNM in order to get nearly uncoupled modal equations.

Micro-crack localization for steel strands using nonlinear vibro-acoustic modulation based on modified signal bispectrum analysis

Under continuous environmental erosion and long-time high tension, fatigue micro-cracks of bridge cables appear frequently to challenge the reliability of cables. The vibro-acoustic modulation analysis is adopted because of its sensitivity to contact-type defects, especially fatigue cracks. In this paper, using multiple sets of low-frequency vibration and high-frequency ultrasonic excitation, the propagation of the ultrasonic wave in the steel strands is studied. Then, the modulated signal bispectrum is proposed to improve the detection reliability of nonlinear modulation components in the nonlinear vibro-acoustic modulation signal of the contact-type defect. The numerical simulations and laboratory tests of straight wire, helical wire, and multi-wire strands are carried out for micro-crack detection to verify the robustness of the proposed method. Compared with the traditional method, the modulated signal bispectrum can still produce clearer peak detection under high noise conditions. The results show that it provides a promising, dynamic and rapid cable defect detection technique.

Influence of the relative stiffness of second-neighbor interactions on chaotic discrete breathers in a square lattice

It is known that the modulational instability of a delocalized nonlinear vibrational mode (DNVM) with frequency outside the phonon band can lead to spontaneous energy localization in a nonlinear lattice on chaotic discrete breathers (CDBs). Considering a β-FPUT square lattice with nearest and next-nearest interactions, the appearance of CDBs is analyzed for different stiffnesses of the first- and second-nearest interactions, k1 and k2, keeping the density of the lattice unchanged. There are two DNVMs in the square lattice with frequencies above the phonon spectrum and both are studied. The appearance of CDBs in the lattice is monitored by calculating the time evolution of the energy localization parameter L and the maximum energy of the particles emax. For solid state physics and materials science, the important range of stiffness parameters is k2<k1, since the stiffness of chemical bonds typically decreases with the distance between atoms. It is found that CDBs form in the square lattice when k2>k1/4. This means that they can form in crystals if the stiffness of the second-neighbor bonds is smaller than that of the first-neighbor bonds, but not too small. Depending on the relation between the anharmonicity parameters of the first- and second-neighbor bonds CDBs can have different polarization.

Fundamental and dipole gap solitons and their dynamics in the cubic–quintic fractional nonlinear Schrödinger model with a [formula omitted]-symmetric lattice

The interplay of two linear controlled terms – fractional diffraction and parity-time (PT) symmetric lattice – gives rise to unique and interesting linear Bloch gap structures within where the nonlinear localized gap modes may exist. In this study, we explore the formation and dynamics of one-dimensional gap solitons in the cubic–quintic physical model combining the fractional diffraction and PT symmetric lattice. Two classes of gap solitons, which we name the fundamental gap solitons and dipole ones, are constructed and their stability regions within the first finite gap of the associated linear Bloch spectrum are identified by means of linear-stability analysis and direct perturbed numerical simulations. We stress that the gap solitons are always unstable under the condition of PT symmetry breaking (the imaginary part of which is above 0.5). The excitations of the stable two classes of gap solitons are also investigated by using the adiabatic variation of the system’s parameters. The results predicted here shed some light on soliton physics in physical systems with combined fractional diffraction and PT symmetric lattice and the competing nonlinearities.

Delocalized nonlinear vibrational modes and discrete breathers in a body centered cubic lattice

Body centered cubic (bcc) lattice with nearest and next-nearest interactions described by the β-FPUT interatomic potential is considered. Exact dynamical solutions in the form of zone-boundary delocalized nonlinear vibrational modes (DNVMs) are analyzed. 31 such solutions are revealed from the analysis of only the symmetry of the bcc lattice. Frequency response of DNVMs for the case of soft- and hard-type anharmonicity is calculated. In the case of hard-type anharmonicity, four DNVMs have frequencies bifurcating from the upper edge of the phonon spectrum and growing with the amplitude. Various quasi-discrete breathers are obtained by superimposing localizing functions upon these DNVMs. Our work demonstrates a practical approach to finding quasi-discrete breathers in higher-dimensional lattices. The obtained spatially localized long-lived vibrational modes inspire the search for various discrete breathers in bcc metals.

Nonlinear neutral modes in zero-mass-flux Plane poiseuille-couette flow

SummaryPlane Poiseuille–Couette flow with zero mean advection velocity, a flow well known to be linearly stable, is analysed at asymptotically large Reynolds number R. A nonlinear neutral mode structure is uncovered which is self-sustained due to the interaction of the viscous wall layers with a predominantly inviscid nonlinear internal critical layer. The O(1) phase speed and wavenumber of the disturbance are computed as functions of the O(R−1/3) wave amplitude, with the formula for the amplitude determined analytically. Two distinct coherent structures are found: one involving a single critical layer and a second with two critical layers. The single critical layer case has a long wavelength/small amplitude limit for which an analytical solution exists, while the dual critical layer structure possesses two solution branches and exists above a threshold amplitude. Both states distort the mean flow by an O(R−1/6) amount, and in the long wavelength limit the distortion becomes comparable in size with the original basic flow. The relevance of this work to the experimental observation of self-sustained states in this flow is discussed.

Multi-harmonic phononic frequency comb generation in capacitive CMOS-MEMS resonators

Phononic frequency combs (PFCs) have emerged as a pivotal technology in precision measurements and advanced signal processing, harnessing the power of discrete, evenly spaced frequency lines. Their expansion into the realm of microelectromechanical systems (MEMS) presents potential opportunities for enhanced control in various applications, from telecommunications to quantum computing. This work presents an experimental study on the generation of PFCs at the fundamental frequency and its higher harmonics in a complementary metal oxide semiconductor based electrostatic MEMS resonator measured under an open-loop configuration. The phenomenon is attributed to nonlinear mode coupling and 1:2 internal resonance between two eigenmodes and their harmonics when the resonator operates in nonlinear resonance. The experimental approach employs a combination of optical and electrical characterization techniques. This study, primarily experimental in nature, provides crucial insights into the behavior of PFCs in MEMS resonators, suggesting possible multi-harmonic internal resonance as the underlying mechanism. These findings contribute significantly to the field of MEMS research, offering potential applications in high-precision frequency control for telecommunications, sensor technology, and quantum computing.

Evolving fatigue damage detection based on multi-component nonlinear modulation mechanism of tuned laser-generated Rayleigh waves

In this paper, we clarify the nonlinear interaction between tuned Rayleigh waves and evolving fatigue damage and achieve evolving fatigue damage detection. Firstly, the impact of laser parameters on Rayleigh waves is investigated to generate stronger Rayleigh waves. Secondly, the multi-component nonlinear modulation mechanism of Rayleigh waves is analyzed using the nonlinear model considering contact and friction. The simulation results indicate that during the plastic deformation stage, waveform distortion is minimal. As microcracks form, the longitudinal wave component becomes dominant in Rayleigh wave nonlinearity due to contact and friction behaviors between microcrack faces, resulting in waveform distortion, harmonic generation, and increased acoustic nonlinearity parameters. However, the shear wave component only exhibits high-frequency filterability without nonlinearity. As macrocracks develop, both longitudinal and shear wave component amplitudes decrease significantly. Finally, an experimental validation is carried out. The experimental results validate that the acoustic nonlinearity parameters first stabilize and then increase significantly from the pristine state to the microcrack propagation, and the indicator can be utilized for fatigue damage detection. The research can provide guidance for nonlinear laser-generated Rayleigh wave detection of fatigue damage.

Modified Mohr–Coulomb criterion for nonlinear strength characteristics of rocks

AbstractThe classical Mohr–Coulomb criterion has an extremely wide range of applications in the field of engineering geology, while it cannot accurately characterize rock failure under high confining stresses. Therefore, a nonlinear modification is implemented based on the Mohr–Coulomb criterion for achieving wider usage in all stress environments. The cohesion and internal friction angle are firstly calculated by regression based on the triaxial experiments of diorite and other typical rocks in different confining stress ranges, and the variation modes of cohesion and internal friction angle are then formalized and integrated into the classical Mohr–Coulomb criterion. Using the experimental results of 435 triaxial compression tests in existing publications, the predictions of the modified criterion are compared with those of the previous nonlinear criteria, which show that the predicted values of the modified criterion are in good agreement with the measured strength. Compared with the previous nonlinear criteria, the modified criterion has a wider application range and always maintains a higher accuracy.

Spectral broadening and nonlinear mode coupling in a gas-filled hollow core capillary

Spectral broadening and nonlinear mode coupling in a gas-filled hollow core capillary

Effects of spot size on the operation mode of GaAs photoconductive semiconductor switch employing extrinsic photoconductivity

To guide the illuminating design to improve the on-state performances of gallium arsenide (GaAs) photoconductive semiconductor switch (PCSS), the effect of spot size on the operation mode of GaAs PCSS based on a semi-insulating wafer with a thickness of 1 mm, triggered by a 1064-nm extrinsic laser beam with the rectangular spot, has been investigated experimentally. It is found that the variation of the spot size in length and width can act on the different parts of the output waveform integrating the characteristics of the linear and nonlinear modes, and then significantly boosts the PCSS toward different operation modes. On this basis, a two-channel model containing the active and passive parts is introduced to interpret the relevant influencing mechanisms. Results indicate that the increased spot length can peak the amplitude of static domains in the active part to enhance the development of the nonlinear switching, while the extended spot width can change the distribution of photogenerated carriers on both parts to facilitate the linear switching and weaken the nonlinear switching, which have been proved by comparing the domain evolutions under different spot sizes.

Nonlinear localized modes in a higher-order anisotropic ferromagnetic nanowire with octupole–dipole interaction

Increasing demand for advanced technologies that depends on magnetic phenomena, understanding and controlling the behavior of discrete breather in ferromagnetic nanowires are crucial for enhancing the efficiency and performance of such devices. The presence of octupole–dipole interactions signifies a unique aspect that could potentially influence the stability and localization of breather excitations. Hence, we adopted a multifaceted approach to investigate the Heisenberg anisotropic ferromagnetic nanowire discrete model with the following interactions: bilinear, octupole–dipole, anisotropy and its higher-order terms. The dynamics is governed by a discrete nonlinear Schrödinger equation (DNLS) arrived with the aid of Holstein–Primakoff transformation. This transformation was facilitated by utilizing the Glauber coherent representation of the boson operators. Subsequently, the dynamical equation is incorporated to the Modulational Instability (MI) analysis which is a systematical gateway to explore the breather excitation in the region of instability influenced by the octupole–dipole interaction coupling parameter. Then, we pictorially demonstrated that the octupole–dipole interaction plays a pivotal role in promoting the localization of discrete breather on the surface of the spin lattice sites in the discrete ferromagnetic nanowire. The energy density distribution also implies that the increase in octupole–dipole interaction results in the highly dense breather localization. The result shows that the increment in the octupole–dipole interaction parameter increases the amplitude of the localized breathers. These discrete breathers could hold immense promise for applications in magnetic storage and Spintronic devices, where maintaining stable localized modes is crucial for the device functionality. Our novelty lies in being pioneers in the exploration of a fully discrete model that encompasses higher-order interactions, such as the octupole–dipole interaction. We already have confirmed the existence of instability region on the discrete spin lattice by incorporating the octupole–dipole interaction [T. Pavithra, L. Kavitha, Prabhu and A. Mani, Modulational instability analysis in an isotropic ferromagnetic nanowire with higher order octopole-dipole interaction, in Nonlinear Dynamics and Applications: Proceedings of the ICNDA 2022 (Springer, 2022), p. 1209], we attempting to explore the generation of discrete breathers in a discrete anisotropic ferromagnetic nanowire. This effectively bridges the gap between theoretical understanding and practical implications, paving the way for innovative advancements in magnetic technology.

Secondary Whistler and Ion-cyclotron Instabilities Driven by Mirror Modes in Galaxy Clusters

Electron cyclotron waves (whistlers) are commonly observed in plasmas near Earth and the solar wind. In the presence of nonlinear mirror modes, bursts of whistlers, usually called lion roars, have been observed within low magnetic field regions associated with these modes. In the intracluster medium (ICM) of galaxy clusters, the excitation of the mirror instability is expected, but it is not yet clear whether electron and ion cyclotron (IC) waves can also be present under conditions where gas pressure dominates over magnetic pressure (high β). In this work, we perform fully kinetic particle-in-cell simulations of a plasma subject to a continuous amplification of the mean magnetic field B (t) to study the nonlinear stages of the mirror instability and the ensuing excitation of whistler and IC waves under ICM conditions. Once mirror modes reach nonlinear amplitudes, both whistler and IC waves start to emerge simultaneously, with subdominant amplitudes, propagating in low- B regions, quasi-parallel to B (t). We show that the underlying source of excitation is the pressure anisotropy of electrons and ions trapped in mirror modes with loss-cone-type distributions. We also observe that IC waves play an essential role in regulating the ion pressure anisotropy at nonlinear stages. We argue that whistler and IC waves are a concomitant feature at late stages of the mirror instability even at high β, and therefore, expected to be present in astrophysical environments like the ICM. We discuss the implications of our results for collisionless heating and dissipation of turbulence in the ICM.

Fundamental frequency and second harmonic mode-locked operations in an erbium-doped fiber laser based on Cr2Sn2Te6 saturable absorbers

With the rapid development of ultrafast photonics, the need for optical modulators is expanding. In particular, the optical modulator with a high damage threshold and the ability to generate multiple soliton states. This has prompted an ongoing search for optical modulators based on two-dimensional (2D) materials. Therefore, in this work, a novel 2D material Cr2Sn2Te6 (CSnT) with modulation depth of 6.17% and saturation intensity of 15.97 MW/cm2 is prepared as saturable absorber (SA). The nonlinear optical absorption characteristics and pulse modulation applications of CSnT are showed in the Erbium-doped fiber laser (EDFL). The fundamental frequency mode-locked operation and the second harmonic mode-locked operation are realized simultaneously in the ring cavity. Moreover, CSnT SA has a high damage threshold and high stability, the damage threshold reaches 467 mW and the signal-to-noise ratio (SNR) is higher than 70 dB. The experimental findings demonstrate that the CSnT-based saturable absorber possesses remarkable nonlinear optical properties, indicating its considerable potential in the realm of ultrafast photonics.

Experimental Investigation of Nonlinear Modulation Characteristics of Mode Localization in Electrostatic Coupled Resonators

This paper focuses on the typical configuration of double ended fixed supported electrostatic coupled resonators in mode localization phenomena, and conducts in-depth research on the influence of nonlinear coefficients in electrostatic coupled resonators on their vibration behavior. Firstly, the designed devices are processed based on the standard process of SOI, and the printed circuit board (PCB) of the test circuit is designed using Altium Designer software to connect external circuits with the tested device. Secondly, the working principles of electrostatic drive and capacitance detection are elucidated, and the mechanical model in the system is transformed into an electrical model through equivalent circuit principles, thereby revealing the influence of parasitic feedthrough capacitance on the detection signal. Finally, an open-loop testing circuit was built in a vacuum environment to test the modal localization characteristics of the electrostatic coupling resonator and the performance of the mode localization acceleration sensor.