Resonance In Nonlinear Systems Research Articles

Multi-mechanical frequency locking mechanism and robustness analysis of MDOF coupled resonator

Mechanical frequency locking can overcome the dependence of peak frequency on the driving amplitude and improve the frequency stability in nonlinear resonant systems. Most research focuses on the single mechanical frequency locking phenomenon, but few studies focus on multi-mechanical frequency locking phenomenon. In this study, the multi-mechanical frequency locking mechanism and robustness analysis of the peak frequency is proposed through multi-mode coupled vibration. Firstly, a multi-degree-of-freedom (MDOF) coupled resonator consisting of a low-frequency resonant element and multiple high-frequency resonant elements is designed, and the vibration energy of the resonant system can be transferred from the low-frequency resonant element driven by a harmonic excitation to the high-frequency resonant elements via the 1:3 internal resonance. Through perturbation and bifurcation analysis, the nonlinear dynamic behaviors of MDOF coupled resonator are analyzed in detail. It is found that the peak frequency of the low-frequency resonant element can remain stable in multiple drive intervals, leading to multi-mechanical frequency locking phenomenon. Then, a 3-DOF magnetic coupled resonant system is experimentally designed to demonstrate the physical conditions and influencing factors of the double mechanical frequency locking phenomenon. Typically, introducing external disturbances, the frequency locking is experimentally demonstrated to improve the robustness of peak frequency. The experimental results show that when the mechanical frequency locking phenomenon occurs, the influence of small external noise disturbance on the peak frequency can be ignored, which greatly improves the frequency stability of nonlinear resonant systems.

Development and possible applications of tunable elements based on composite materials

This work is devoted to the synthesis of nano-sized weakly agglomerated magnetic particles with different crystal structures, which were used in the development of nonlinear composite structures, in particular, nonlinear resonant microwave elements, left-handed media, and nonlinear magnetoelectric systems.

Double mechanical frequencies locking phenomenon in a piezoelectric driven 3-DOF magnetic coupling resonator

Frequency locking phenomenon induced by modal coupling can effectively overcome the dependence of peak frequency on driving strength in nonlinear resonant systems and improve the stability of peak frequency, which has a fundamental role in improving the dynamic performance of resonant devices. Most research has been focused on a single mechanical frequency locking, but less research has been done on the double frequencies locking phenomenon. In this study, the double frequencies locking phenomenon is first proposed through a 1:3:3 internal resonance of a three degrees of freedom (3-DOF) magnetic coupled resonant system driven by piezoelectricity. Firstly, the resonator is designed using three cantilever beams, permanent magnets and piezoelectric actuator. The nonlinear governing equations, representing the first, second and third modes, are obtained by a 3-DOF lumped parameter model of the resonator. The double frequencies locking phenomenon in a 3-DOF resonant system is measured experimentally and studied theoretically. Both theoretical predictions and experimental results show that with the increase of driving force, the first frequency locking phenomenon occurs due to the coupling vibration between the first and second modes. Interestingly, as the drive force continues to increase, the first frequency locking switches to the second frequency locking, accompanied by the transfer of vibration energy from the first mode to the third mode. Typically, the physical conditions of the first and second frequencies locking are deduced theoretically. Besides, a synchronous linear mass sensing scheme for double analytes is proposed by using double frequencies locking phenomenon. By measuring the shifts of the first and second locking frequencies, the mass detection of the double analytes can be carried out simultaneously.

Resonances and Phase Locking Phenomena for Foliation Preserving Torus Maps

It is well known for experts that resonances in nonlinear systems lead to new invariant objects that lead to new behaviors. The goal of this paper is to study the invariant sets generated by resonances under foliation preserving torus maps. That is torus which preserve a foliation of irrational lines $L_{\theta_{0}}=\{\theta_{0}+\Omega t | t\in\mathbb{R}\}\subset\mathbb{T}^{d}$. Foliation preserving maps appear naturally as reparametrization of linear flows in the torus and also play an important role in several applications involving coupled oscillators, delay equations, resonators with moving walls, etc. The invariant objects we find here, lead to predictions on the behavior of these models. Since the results of this paper are meant to be applied for other problems, we have developed very quantitative results giving very explicit descriptions of the phenomena and the invariant objects that control them. The structure of the phase locking regions for foliation preserving maps is very different than for generic maps of the torus. Indeed, for the sake of completeness, we have developed similar analysis for the case of generic maps of the torus and shown that the objects that appear in foliation preserving maps are quantitatively and qualitatively different from those of generic torus maps. This has consequences in applications.

Observer-based formation tracking control for feedforward nonlinear multi-agent systems with dead-zone input

In this paper, the formation tracking control problem is addressed for feedforward nonlinear multi-agent systems with dead-zone input. For each follower, a dynamic observer is constructed by the output information of itself and its neighbours. Based on the observer, a distributed control protocol is designed with a bounded time-varying gain to achieve the formation tracking between each follower and the leader. It should be pointed out that the control protocol has fewer communication variables and simpler structure, while improving the convergence speed and transient performance of system. In addition, the results are extended to the feedback nonlinear multi-agent systems. Finally, a practical example is performed for nonlinear liquid level resonant circuit system to demonstrate the feasibility of the control protocol proposed.

Comparative study of modal decomposition and dynamic equation reconstruction in data-driven modeling

Due to the increasing complexity of dynamic systems, it is increasingly difficult for traditional mathematical methods to meet the modeling requirements of complex dynamic systems. With the continuous innovation of computer and big data technologies, massive data can be easily obtained and stored. Therefore, studies of dynamic system modeling through data-driven approaches have attracted more and more researchers’ attention. This paper compares the dynamic mode decomposition method and dynamic equation reconstruction. Taking Lorenz and nonlinear Helmholtz resonant systems as examples, the two methods show the ability to reconstruct and describe the evolution characteristics of the dynamic system. Specifically, the dynamic mode decomposition method can describe the characteristics of the dynamic system more intuitively; however, it cannot provide physical insights. On the other hand, the discovery of dynamic equations from data can more accurately express the physical evolution characteristics of the dynamic system; however, it is easily affected by random noise. Because the dynamic mode decomposition method can obtain a reduced-order model, which can not only retain useful information of the original data but also reduce the noise disturbances, it can effectively improve the noise attenuation and finally reconstructions of differential dynamical equations.

Stochastic and vibrational resonance in complex networks of neurons.

The concept of resonance in nonlinear systems is crucial and traditionally refers to a specific realization of maximum response provoked by a particular external perturbation. Depending on the system and the nature of perturbation, many different resonance types have been identified in various fields of science. A prominent example is in neuroscience where it has been widely accepted that a neural system may exhibit resonances at microscopic, mesoscopic and macroscopic scales and benefit from such resonances in various tasks. In this context, the two well-known forms are stochastic and vibrational resonance phenomena which manifest that detection and propagation of a feeble information signal in neural structures can be enhanced by additional perturbations via these two resonance mechanisms. Given the importance of network architecture in proper functioning of the nervous system, we here present a review of recent studies on stochastic and vibrational resonance phenomena in neuronal media, focusing mainly on their emergence in complex networks of neurons as well as in simple network structures that represent local behaviours of neuron communities. From this perspective, we aim to provide a secure guide by including theoretical and experimental approaches that analyse in detail possible reasons and necessary conditions for the appearance of stochastic resonance and vibrational resonance in neural systems. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 2)'.

Dynamical Systems, Celestial Mechanics, and Music: Pythagoras Revisited

Gioseffo Zarlino reintroduced the Pythagorean paradigm into Renaissance musical theory. In a similar fashion, Nicolaus Copernicus, Galileo Galilei, Johannes Kepler, and Isaac Newton reinvigorated Pythagorean ideas in celestial mechanics; Kepler and Newton explicitly invoked musical principles. Today, the theory of dynamical systems allows us to describe very different applications of physics, from the orbits of asteroids in the Solar System to the pitch of complex sounds. Our aim in this text is to review the overarching aims of our research in this field over the past quarter of a century. We demonstrate with a combination of dynamical systems theory and music theory the thread running from Pythagoras to Zarlino that allowed the latter to construct musical scales using the ideas of proportion known to the former, and we discuss how the modern theory of dynamical systems, with the study of resonances in nonlinear systems, returns to Pythagorean ideas of a Musica Universalis.

Feedthrough parasitic nonlinear resonance in micromechanical oscillators

Mechanical oscillators incorporating miniaturized structures to transduce mesoscopic motion as readable electrical signals are often limited by the feedthrough effect (FE) because oscillation features are corrupted. The knowledge of FE parasitic resonance hitherto is only demonstrated in the linear regime. Herein, we reveal the nature of phenomenological FE parasitic nonlinear resonance. An inverse distortion in the spectra resulting from the superposition of Duffing nonlinearity and the FE is observed and modeled with physical insight into the governing parameters. We find that the manipulation of electromechanical coupling of the oscillator can fully de-embed the FE, while the device oscillates at the nonlinear bifurcation point under phase control. The generic nature of our model indicates that similar dynamic behaviors will occur for the nonlinear resonant systems containing the FE, regardless of transducing techniques. The strategy of on-chip FE control in our capacitive platform is scalable and can be suitably transformed for application in oscillators employing alternative transducers.

Active control for acoustic wave propagation in nonlinear diatomic acoustic metamaterials

Wave propagation through nonlinear acoustic metamaterials has generated numerous scientific interests for their enormous potential in practical applications these years. This study focuses on the effects of nonlinearity on the band properties of diatomic mass-in-mass chain with active control. By applying the Lindestedt–Poincaré (L–P) perturbation method, analytical dispersion relations of the linear and nonlinear diatomic mass-in-mass system have been established and investigated by numerical simulation. Different from the monatomic mass-in-mass chain, this two mass-in-mass units forming a unit cell of the periodic structure results in four branches of the dispersion relation. The effects of nonlinearity on the band gaps of the system have been exhaustively illustrated. By only tuning the nonlinear constitutive relation parameter of the spring, the fourth branch and the third gap are found to be more sensitive compared to the other branches and gaps. It is concluded that closing and re-opening of the band-folding-induced gap in this nonlinear system is still possible. Here, a piezoelectric spring model is applied to the diatomic mass-in-mass to make the system available for wider applications. With the negative proportional control, a new stop band is generated which can be also captured in the monatomic nonlinear system. The new results here will help better analyze the band gap properties in nonlinear mechanical metamaterials and emphasize the great potentials of the topological analysis of such a nonlinear local resonance system that induces band-folding-induced band gaps.

Melonic Turbulence

We propose a new application of random tensor theory to studies of non-linear random flows in many variables. Our focus is on non-linear resonant systems which often emerge as weakly non-linear approximations to problems whose linearized perturbations possess highly resonant spectra of frequencies (non-linear Schr\"odinger equations for Bose-Einstein condensates in harmonic traps, dynamics in Anti-de Sitter spacetimes, etc). We perform Gaussian averaging both for the tensor coupling between modes and for the initial conditions. In the limit when the initial configuration has many modes excited, we prove that there is a leading regime of perturbation theory governed by the melonic graphs of random tensor theory. Restricting the flow equation to the corresponding melonic approximation, we show that at least during a finite time interval, the initial excitation spreads over more modes, as expected in a turbulent cascade. We call this phenomenon melonic turbulence.

Magnetic Anomaly Signal Detection Using Parallel Monostable Stochastic Resonance System

The nonlinear stochastic resonance (SR) system possesses the ability of taking advantage of noise to enhance the weak signal when the SR system, signal and noise reach to the matching relation. It provides an effective approach to detect the weak magnetic anomaly signal in low signal-to-noise ratio. However, in practical applications, the measured magnetic anomaly signal may be a peak signal, a trough signal, or a combination of the two due to the uncertainty of magnetic target orientation. Hence it is difficult to maintain a good detection performance with single SR system because the SR system output is directly influenced by signal waveforms. Aiming to this, a new strategy using the parallel monostable SR (PMSR) system is proposed, which can ensure the good detection performance regardless of a peak signal, a trough signal, or a combination of the two. Besides, we take the kurtosis index as the criterion and search the optimal system parameters in SR system. The simulation and experiment results indicate its availability, validity and that it can achieve a good detection performance in different waveforms. It can be expected to be widely used in the field of magnetic anomaly detection with PMSR system.

Two infinite families of resonant solutions for the Gross-Pitaevskii equation

We consider the two-dimensional Gross-Pitaevskii equation describing a Bose-Einstein condensate in an isotropic harmonic trap. In the small coupling regime, this equation is accurately approximated over long times by the corresponding nonlinear resonant system whose structure is determined by the fully resonant spectrum of the linearized problem. We focus on two types of consistent truncations of this resonant system: first, to sets of modes of fixed angular momentum, and second, to excited Landau levels. Each of these truncations admits a set of explicit analytic solutions with initial conditions parametrized by three complex numbers. Viewed in position space, the fixed angular momentum solutions describe modulated oscillations of dark rings, while the excited Landau level solutions describe modulated precession of small arrays of vortices and antivortices. We place our findings in the context of similar results for other spatially confined nonlinear Hamiltonian systems in recent literature.

Stochastic resonance characteristic analysis of new potential function under Levy noise and bearing fault detection

Based on the output saturation of classcial bistable stochastic resonance (CBSR), a new type of piecewise nonlinear bistable stochastic resonance (PNBSR) system is constructed. The mean signal-to-noise ratio gain is regarded as an index to measure the stochastic resonance phenomenon. The laws for the resonant output of piecewise nonlinear bistable system governed by l, c, a, b and D of Levy noise are explored under different characteristic index α and symmetry parameter β of Levy noise. The results show that the output of PNBSR system has increased 4 dB by comparing with the output signal-to-noise ratio of CBSR system. And the stochastic resonance phenomenon can be induced by adjusting the piecewise nonlinear system's parameters under any α or β of Levy noise. The interval of the parameters of system which induces good stochastic resonance is roughly the same. And the output signal waveform of resonance is very similar to the input signal waveform, which has some reference value for the signal recovery. Moreover, we can find the good stochastic resonance interval of the system parameters do not change with D of Levy noise under the different noise intensity D of Levy noise. On the basis of this, adjusting the intensity amplification factor D of Levy noise, which induces good stochastic resonance, and the interval does not change with α or β. At last, the piecewise nonlinear bistable system is applied to detect bearing fault signals, which achieves better performance compared with the classical bistable system.

Resonance Principle for the Design of Flapping Wing Micro Air Vehicles

Achieving resonance in flapping wings has been recognized as one of the most important principles to enhance power efficiency, lift generation, and flight control performance of high-frequency flapping wing micro air vehicles (MAVs). Most work on the development of such vehicles have attempted to achieve wing flapping resonance. However, the theoretical understanding of its effects on the response and energetics of flapping motion has lagged behind, leading to suboptimal design decisions and misinterpretations of experimental results. In this work, we systematically model the dynamics of flapping wing as a forced nonlinear resonant system, using both nonlinear perturbation method and linear approximation approach. We derived an analytic solution for steady-state flapping amplitude, energetics, and characteristic frequencies including natural frequency, damped natural frequency, and peak frequency. Our results showed that both aerodynamic lift and power efficiency are maximized by driving the wing at natural frequency, instead of other frequencies. Interestingly, the flapping velocity is maximized at natural frequency as well, which can lead to an easy experimental approach to identify natural frequency and validate the resonance design. Our models and analysis were validated with both simulations and experiments on ten different wings mounted a direct-motor-drive flapping wing MAV. The result can serve as a systematic design principle and guidance in the interpretations of empirical results.

Nonstationary weak signal detection based on normalization stochastic resonance with varying parameters

The nonlinear stochastic resonance system possesses the ability of taking advantage of background noise to enhance the weak signal. It provides a new approach to detect the weak signal embedded with heavy noise. This study proposes a new varying parameter stochastic resonance employing the fourth-order Runge–Kutta numerical method as well as the normalized transformation of a bistable stochastic resonance system. The model performs well in the detection of a time-varying signal with background noise for denoising and signal recovery. We take the fitness coefficient and cross-correlation coefficient as the criteria and analyze the influence of different parameters. The simulating results indicate its availability, validity and that it generates a better performance than the traditional stochastic resonance. The method develops the area of time-varying signal detection with stochastic resonance and presents new strategy for detection and denoising of a time-varying signal. It can be expected to be widely used in the areas of aperiodic signal processing, radar communication, etc.

Nonstationary energy localization vs conventional stationary localization in weakly coupled nonlinear oscillators

In this paper we study the effect of nonstationary energy localization in a nonlinear conservative resonant system of two weakly coupled oscillators. This effect is alternative to the well-known stationary energy localization associated with the existence of localized normal modes and resulting from a local topological transformation of the phase portraits of the system. In this work we show that nonstationary energy localization results from a global transformation of the phase portrait. A key to solving the problem is the introduction of the concept of limiting phase trajectories (LPTs) corresponding to maximum possible energy exchange between the oscillators. We present two scenarios of nonstationary energy localization under the condition of 1:1 resonance. It is demonstrated that the conditions of nonstationary localization determine the conditions of efficient targeted energy transfer in a generating dynamical system. A possible extension to multi-particle systems is briefly discussed.

Vibrational Resonance in a Duffing System with a Generalized Delayed Feedback

We investigate the vibrational resonance in the Duffing system with different kinds of delayed feedback. Our approach is to consider the delayed feedback as a generalized delayed feedback in a fractional-order differential version. For three special cases, the generalized delayed feedback corresponds to displacement delayed feedback, velocity delayed feedback, and acceleration delayed feedback respectively. At first, based on the vibrational mechanism, the approximate solution of the system is obtained. Then, we give conditions for all resonance patterns. The the oretical predictions are verified by numerical simulations. Furthermore, the theoretical results are in good agreement with the numerical simulations. Since the delayed feedback is in a generalized form, our results can be regarded as universal for the vibrational resonance in nonlinear systems with different kinds of delayed feedback.

Effects of Barrier Parameter to Stochastic Resonance Signal-to-noise Ratio in Feature Extraction

Effects of barrier parameter a to output signal-to-noise ratio (SNR) of bistable stochastic resonance in feature extraction was investigated in this paper. Barrier parameter a was changed with other systematic parameters fixed. The relationship between parameter a and the output SNR of non-linear stochastic resonance system was studied. This research provided us a novel way to extract the features using the non-linear stochastic resonance. DOI: http://dx.doi.org/10.11591/telkomnika.v11i12.3724 Full Text: PDF

A dominant role for mechanical resonance in physiological finger tremor revealed by selective minimization of voluntary drive and movement

There is a debate in the literature about whether the low- and high-frequency peaks of physiological finger tremor are caused by resonance or central drive. One way to address this issue is to examine the consequences of eliminating, as far as possible, the resonant properties or the voluntary drive. To study the effect of minimizing resonance, finger tremor was recorded under isometric conditions and compared with normal isotonic tremor. To minimize central drive, finger tremor was generated artificially by broad-band electrical stimulation. When resonance was minimized, tremor size declined almost monotonically with increasing frequency. There was no consistent large peak at a frequency characteristic of tremor. Although there was sometimes a peak around the tremor frequency during some isometric conditions, it was extremely small and variable; therefore, any contribution of central drive was minimal. In contrast, there was always a prominent peak in the isotonic frequency spectra. Resonance was, therefore, necessary to produce the characteristic tremor peaks. When central drive was minimized by replacing voluntary muscle activation with artificial stimulation, a realistic tremor spectrum was observed. Central drive is, therefore, not required to generate a characteristic physiological tremor spectrum. In addition, regardless of the nature of the driving input (voluntary or artificial), increasing the size of the input considerably reduced isotonic tremor frequency. We attribute the frequency reduction to a movement-related thixotropic change in muscle stiffness. From these results we conclude that physiological finger tremor across a large range of frequencies is produced by natural broad-band forcing of a nonlinear resonant system, and that synchronous central input is not required.