Nonlinear Resonance Phenomena Research Articles

Nonlinear dynamics of a horizontal rotor with asymmetric magnetic supports

In this paper, a horizontal rotor with passive anisotropic asymmetric magnetic bearings nonlinear model is developed. The model is based on the experimental evidence of an educational demonstrator, a powerful benchmark which highlights features of rotordynamics systems. An experimental setup is developed using laser sensors to track the displacements of the rotor and tachometer to record the rotor angular speed. Several test campaigns are performed with different initial conditions to characterise the educational demonstrator. The first test campaign is oriented to identify the system structural properties, while the second one is focused on the nonlinearity identification. The rotor modelling is divided into three stages with progressive increase in complexity, starting from the overall linear structural behaviour to the nonlinear subharmonic resonance phenomenon. The model is developed with a generalised approach suitable for a wide range of operating conditions and rotor configurations. Progressively, the developed rotor models are compared to highlight their limitations and advantages. Finally, rotor trajectory analysis and time-frequency analysis are used in the numerical-to-experimental comparison.

Nonlinear vibrations of a linkage mechanism: theory and experiment

Linkage mechanisms have a wide range of applications in truck-mounted cranes, folding mechanical arms of transport vehicles and so on due to their simple and effective design. Vibration problems in the linkage mechanisms seriously affect the stability and reliability of crane machines or transport vehicles. Therefore, in this study, an effective nonlinear dynamic model of a linkage mechanism with spring-damping systems is developed and experimentally validated to explore the nonlinear vibration properties. The geometric nonlinearity of the linkage mechanism is considered and the Lagrange equation is employed to establish the nonlinear dynamic equations. Based on the harmonic balance method (HBM), the nonlinear vibration properties of the structural system are analyzed. The nonlinear primary resonance response of the linkage mechanism calculated from the HBM is compared with those of the Runge–Kutta method and the experimental result, which validates the effectiveness and accuracy of the proposed nonlinear dynamic model. The influences of some parameters, such as the spring stiffness, the lengths of connecting rods, and the installation positions of the spring-damping system, on the nonlinear vibration characteristic of the linkage mechanism are analyzed. Based on the numerical results, the folding mechanical arms or truck-mounted cranes can be conveniently designed to avoid irreparable damage to the structure caused by nonlinear primary resonance and instability phenomena, which provides new insight into the stability analysis and design of various linkage mechanisms required in engineering applications.

Auto-parametric resonance of flexible viscoelastic beams under interaction between longitudinal and transverse modes

For a system encompassing subsystems, dynamic modal interaction may experience unanticipated vibrations among subsystems that are excited indirectly. This paper focuses attention to a nonlinear auto-parametric resonance phenomenon of a flexible beam introduced via modal coupling of longitudinal forced resonance and transverse parametric resonance under longitudinal pulsatile excitation. The coupled resonance equations are first modeled based on the Euler-Bernoulli beam theory by incorporating nonlinear curvature, von Kármán's nonlinear strain-displacement relationship and the flexible beam model characterizing nonlinear inertia and displacement interaction. The Galerkin truncation-incremental harmonic balance method (IHBM) is then introduced to obtain both the stability boundary and amplitude-frequency bifurcation maps. Through the corresponding relationship between stability boundary and bifurcation diagram, the energy exchange between two modes is identified as well as the instability region and bifurcation topology can be tuned. It is found that the modal interaction greatly widens the instability region with at most two resonance bands of lateral resonance. By tuning structural and foundation damping, resonance can be controlled to occur only in one frequency band of the pair as desired. This research provides guidance to design broadband energy harvesting or vibration suppression equipment with tunable working frequency band for using the strategy of auto-parametric resonance.

Nonlinear optomechanical resonance entering a self-organized energy transfer pattern

The energy transfer between different subsystems or different vibration modes is always one of the most interested problems in the study of the resonance phenomena in coupled nonlinear dynamical systems. With an optomechanical system operating in the regime of unresolved sideband, where its mechanical frequency is lower than the cavity field damping rate, we illustrate the existence of a special nonlinear resonance phenomenon. This type of previously unknown resonance manifests an organized pattern of the coupled cavity field and mechanical oscillation, so that the cavity field precisely pushes the mechanical oscillator within an appropriate small time window in each mechanical oscillation period and the mechanical energy will increase by a jump of almost fixed amount after each oscillation cycle. The scenario is realized at a resonance point where the frequency difference of two driving fields matches the mechanical frequency of the system, and this condition of drive-frequency match is found to trigger a mechanism to lock the two subsystems of an unresolved-sideband optomechanical system into a highly ordered energy transfer as the above mentioned. Due to a significantly enhanced nonlinearity in the vicinity of the resonance point, optical frequency combs can be generated under pump powers of thousand times lower, as compared to the use of a single-tone driving field for the purpose. An unresolved sideband system under the drives without satisfying the resonance condition also demonstrates other interesting dynamical behaviors. Most of all, by providing a realistic picture for the nonlinear optomechanical dynamics in unresolved sideband regime, our study points to a direction to observe novel dynamical phenomena and realize other applications with the systems of less technical restrictions.

Nonlinear resonant phenomena in the secondary suspension system of the vehicle cab research

The issue of creating a methodology for analyzing nonlinear phenomena in the occurrence of spatial resonances in the truck cab suspension system is solved in research. The study purpose is the creating of spatial vibrations mathematical model of the cab on the suspension with regard to the car frame and changing the differential equations of the model dynamics to a form that allows visually analyze the nonlinear resonant phenomena arising in the system to research oscillations instability in resonance conditions. The transformation of the oscillation equations to a quasi-linear form is performed using the replacement of the trigonometric functions of Euler angles by the first two terms of their power series expansions. Because of the research, unfavorable frequency ratios for external and internal resonances in the cab suspension system were obtained. Guides are given to reduce the cab on the suspension rocking amplitudes when spatial nonlinear resonances occur. Keywords: nonlinear resonance, secondary suspension system, car cab suspension, mathematical model, quasi-linear view, vibration stability

Diagnosis of Chaotic Ferroresonance Phenomena Using Deep Learning

Ferroresonance is a non-linear and dangerous resonance phenomenon that can affect power networks and damage electrical equipment. The ferroresonance phenomenon is examined by dividing it into classes, with chaotic ferroresonance being the most dangerous type that causes overvoltage’s. Detecting chaotic ferroresonance in a short period of time is of great importance in terms of taking measures and reducing equipment damage. In this study, we explored the application of deep convolutional neural networks (DCNNs) for the identification and classification of chaotic ferroresonance phenomena. Two pre-trained AlexNet models were adapted using transfer learning to perform these tasks. The first model was utilized to identify chaotic ferroresonance, while the second was employed to distinguish between different subtypes of chaotic ferroresonance by dividing voltage curve graphs into different periods and shapes. The training and testing of both DCNN models were conducted using snapshot images extracted from the voltage curves of all phase voltages. The results of the experiments showed high accuracy in both the identification and classification of chaotic ferroresonance phenomena.

Internal resonance of generalized suspension bridge model considering torsional-vertical vibration

Non-linear vibration is a crucial issue for suspension bridge due to its slenderness and low damping. This study proposes a fully mathematical model of a suspension bridge considering arbitrary inclined angle of main cables by using principle of Hamilton. This is the first mathematical model that is capable of analyzing the nonlinear coupled vertical-torsional vibration of generalized bridge configurations with independently considering the motion of main cables and girder. Galerkin method is adopted for the discretization of the model, and the multi-scale method is employed to acquire modulation equations. The accuracy of the non-linear vibration obtained from proposed mathematical model are validated by a FE model. The internal resonance analysis for such a suspension bridge with spatial layout of main cables between vertical and torsional modes with frequency ratio of 2:1, and between two torsional modes respectively dominated by deck and main cables with frequency ratio of 1:1 are successively investigated in the first time within a narrow detuning frequency domain. The phenomenon of nonlinear resonance was discussed with different inclination angle of main cables and detuning parameters. Some novel and practical conclusions were obtained. Results show that 2:1 internal resonance between vertical and torsional modes may be induced only when two forces with frequencies corresponding to involved modes are applied simultaneously. In addition, significant 1:1 internal resonance is observed, with inducing Hopf bifurcation. Understanding such possible situations is favorable to the design of bridge structure.

An arbitrary Lagrangian-Eulerian method for nonlinear structural-acoustic interaction of hyperelastic solid and compressible viscous fluid

This paper develops a numerical method for predicting the nonlinear structural-acoustic interactions of a hyperelastic solid immersed in an unbounded compressible viscous fluid. The nonlinear acoustic wave model with perfectly matched layers adopted is proposed in an arbitrary Lagrangian-Eulerian frame to absorb acoustic waves at truncated fluid boundaries, whereas the geometric and material nonlinearities of the hyperelastic solid are characterized in a total Lagrangian form. Both the fluid and the solid are discretized by the finite element method, and both models are coupled together by the momentum equilibrium and kinematic compatibility on the common interface. The convergence and accuracy of the proposed method are assessed by comparing the computed results with existing solutions. A hyperelastic ring in a viscous fluid is analyzed to provide physical insights into the nonlinear dynamics behaviors of the coupled hyperelastic solid and acoustic fluid system. The hyperelastic ring in acoustic fluid exhibits deformation instability under a uniform distributed excitation load, producing a series of subharmonics, ultrasubharmonics, and superharmonics in the structural and acoustic responses. Nonlinear internal resonance and bifurcation phenomena in the vibration of the ring and the acoustic waves of the fluid are also discussed.

The investigation of ferro resonance voltage fluctuation considering load types and damping factors

Abstract Background and Objectives: The occurrence of ferro resonance is a very complex phenomenon in power systems. In circuits including inductors and capacitors, if the value of zc = zl becomes, an intensification or resonance state is created in the circuit. In circuits and networks that have saturated inductors, nonlinear resonance or ferro resonance is created, Ferro resonance is a nonlinear resonance phenomenon that occurs between the network capacitor and the nonlinear inductance of the transformer during saturation. Ferro resonance at the main frequency and higher frequencies cause insulation problems and at low frequencies causes thermal problems. In the event of this phenomenon, the voltage range increases to a considerable extent and may also be distorted, which due to the structure of the winding connections in transformers can cause damage to electrical installations. Methods: In this paper, we study and simulate the phenomenon of ferro resonance and provide solutions such as using high resistance on the primary side of the transformer and the combination of resistors and inductors on the secondary side of the transformer to damp the ferro resonance in MATLAB software. Furthermore, the utilization of surge arresters is discussed in detail. Results: It is illuminated that utilization of suggested approaches has an acceptable reduction rate on the damping of ferro resonance fluctuations. Especially when using resistor and inductor simultaneously the fluctuations reduce and ferro resonance is damped immediately. Conclusion: It is important to pay attention to the transformer and the amount of capacitors in transmission lines, especially cable lines. Furthermore, asymmetric switching is another important factor. High resistance can reduce fluctuations but causes power losses in the circuit but using Resistance and inductor structure are able to create acceptable damping and reduce fluctuations without loss problems. The presence of a surge arrester can also reduce the overvoltage caused by ferro resonance to an appropriate level.

Hybrid nonlinear resonance in Hamiltonian systems.

An electronic system in an atom can be considered Hamiltonian only at times shorter than the spontaneous relaxation time. However, this time is sufficient for resonant action on the electronic system and for the implementation of the resonance inherent in Hamiltonian systems. In practice, there may be a case when it is expedient to use a hybrid approach to study nonlinear resonance, in which the classical theory can be used to calculate the action-dependent nonlinear resonance frequency, and the quantum theory can be used to calculate its correction. The use of such a hybrid approach becomes necessary when the resonant value of the action does not exceed Planck's constant many times. It is shown in the work that if the external electromagnetic field has the form of a periodic series of light pulses with a high duty cycle, then the phenomenon of nonlinear hybrid resonance leads to the appearance of a line in the low-frequency region of the electronic spectrum. The broadening of this line is determined using the rms quantum fluctuations.

Optimal resonance response of nonlinear system excited by nonlinear frequency modulation signal

Nonlinear frequency modulation (NLFM) signal is widely used in radar, communication and signal processing. The response of nonlinear system excited by this kind of signal has rich information. At the same time, enhancing different types of signals by resonance phenomenon has unique advantages in the field of signal processing. Compared with other signal processing methods, such as empirical mode decomposition, variational mode decomposition, wavelet transform, signal filtering, etc., this kind of method can not only enhance the signal, but also effectively suppress the interference noise. Therefore, it has certain significance to study the nonlinear system optimal response excited by different NLFM signals and enhance the NLFM signal through resonance phenomenon. In this paper, what is mainly studied is the nonlinear system resonance phenomenon excited by different NLFM signals, which is different from in previous studies. Firstly, a real-time scale transformation method is proposed to process the NLFM signals, and its basic principle is to match different NLFM signals by real-time scale coefficients and system parameters. The signal frequency at each time corresponds to the coefficients with different scales and system parameters, thereby realizing the optimal resonance response at each time. In order to describe the optimal resonance response excited by the NLFM signal more accurately, unlike the traditional spectral amplification factor, the real-time spectral amplification factor is introduced as an evaluation index. Then, the influence of system parameters on the optimal system resonance response is discussed, and the optimal resonance region is obtained, which means that the optimal resonance response can be achieved by selecting the parameters in a reasonable range. This method not only greatly enhances the signal characteristics, but also maintains the continuity of signal time-frequency characteristics. Finally, the real-time scale transformation method is compared with the general scale transformation method, showing the superiority of the proposed method in processing NLFM signal. The method and the results of this paper show some potential in dealing with complex NLFM, which provides a reference for NLFM signal enhancement and detection, and has a certain practical significance in signal enhancement. Furthermore, the relevant influence law of the system optimal response excited by the NLFM signal is given, which has a certain reference value for studying the system dynamic behavior under different signal excitations.

Synchronous identification and successive detection of multiple traces with tunable coupling oscillators

Trace identification and detection of toxic, flammable, explosive substances, pollen allergens, as well as biological viruses is crucial for public health and safety, chemic production, and scientific research, etc. A single trace is detectable via various nonlinear resonance phenomena, while quantitative detection of two traces is physically limited by the trace ratio even with localized oscillators. We have discovered that, in internal resonance systems, response amplitude of any coupled mode physically reflects multi-modal features. In case of multi-mode coupled duffing oscillators, peaks and jumps in response amplitudes show great potential in synchronous identification and successive detection for multiple traces. With tunable magnetically coupled duffing oscillators, multiple traces are both experimentally and theoretically demonstrated to be synchronously identified and successively detected with multiplied frequency shifts. Measurement ranges and manufacturing errors are adjustable or compensable respectively only through turning the coupling strength. This work not only opens a new avenue in applications of nonlinear resonators, but also provides a new metrological scheme for synchronous identification and successive detection of multiple traces.

An instantaneous-baseline multi-indicial nonlinear ultrasonic resonance spectral correlation technique for fatigue crack detection and quantification

This article presents an instantaneous-baseline multi-indicial nonlinear ultrasonic resonance spectral correlation technique for fatigue crack detection and quantification. A reduced-order nonlinear oscillator model is tailored to illuminate the contact acoustic nonlinearity (CAN) and the nonlinear resonance phenomena. The analytical formulation considers the rough surface condition of the fatigue cracks, with a crack open–close transitional range for the effective modeling of the variable-stiffness nonlinear mechanism. Multiple damage indices (DIs) associated with the degree of nonlinearity of the interrogated structures are then proposed by correlating the ultrasonic resonance spectra. Three perspectives of the nonlinear resonance phenomena are investigated to detect and monitor the fatigue crack growth: (1) time-history dependence, which evolves different resonance states depending on the loading history; (2) amplitude dependence, which renders significantly different nonlinear responses under various levels of excitation amplitudes; (3) breakage of superposition, which effectively distinguishes nonlinear resonant responses from the linear counterparts. These DIs are established using instantaneous baselines, facilitating the fatigue damage monitoring without the prior knowledge of a pristine structure. Fatigue tests on a thin aluminum plate with a rivet hole are conducted to induce fatigue cracks in the specimen. The experimental results demonstrate that the proposed technique shows remarkable sensitivity to the nucleation and growth of the fatigue cracks. This paper differs from the existing literature on nonlinear resonance-based techniques in that it focuses on the resonance phenomenon aroused by the contact acoustic nonlinearity from localized fatigue cracks, rather than the diffused material nonlinearity. The novelty of the paper resides in the establishment of an instantaneous baseline technique utilizing the nonlinear resonance features without the need of referring to a pristine baseline situation. The paper finishes with discussion, concluding remarks, and suggestions for future work.

Dynamical behaviors and oblique resonant nonlinear waves with dual-power law nonlinearity and conformable temporal evolution

<p style='text-indent:20px;'>In this article, the oblique resonant traveling waves and dynamical behaviors of (2+1)-dimensional Nonlinear Schrödinger equation along with dual-power law nonlinearity, and fractal conformable temporal evolution are reported. The considered equation is converted to an ordinary differential equation by taking the traveling variable wave transform and properties of Khalil's conformable derivative into account. The modified Kudryashov method is implemented to divulge the oblique resonant traveling wave of such an equation. It is found that the obliqueness is only affected on width, but not on amplitude and phase patriots of resonant nonlinear propagating wave dynamics. The research outcomes are very helpful for analyzing the obliquely propagating nonlinear resonant wave phenomena and their dynamical behaviors in several nonlinear systems having Madelung fluids and optical bullets.

Nonlinear resonances phenomena in a modified Josephson junction model

In this paper, the equivalent circuit of the non-autonomous Josephson junction (JJ) is presented and the effect of the proper frequency on the phase ϕ is studied. We also study nonlinear resonance phenomena in the oscillations of a modified Josephson junction (MJJ). These oscillations are probed through a system of nonlinear differential equations and the multiple time scale method is employed to investigate all different types of resonance that occur. The results of primary, superharmonic and subharmonic resonances are obtained analytically. We show that the system exhibits hardening and softening behaviors, as well as hysteresis and amplitude hopping phenomena in primary and superharmonic resonances, and only the hysteresis phenomenon in subharmonic resonance. In addition, the stabilities and the steady state solutions in each type of resonances are kindly evaluated. The number of equilibrium points that evolve with time and their stabilities are also studied. Finally, the equations of motion are numerically integrated to check the correctness of analytical calculations. We further show that the dynamics of the MJJ is strongly influenced by its parameters.

Comparison of Nonlinear Wave-Loading Models on Rigid Cylinders in Regular Waves

Monopiles able to support very large offshore wind turbines are slender structures susceptible to nonlinear resonant phenomena. With the aim to better understand and model the wave-loading on these structures in very steep waves where ringing occurs and the numerical wave-loading models tend to lose validity, this study investigates the distinct influences of nonlinearities in the wave kinematics and in the hydrodynamic loading models. Six wave kinematics from linear to fully nonlinear are modelled in combination with four hydrodynamic loading models from three theories, assessing the effects of both types of nonlinearities and the wave conditions where each type has stronger influence. The main findings include that the nonlinearities in the wave kinematics have stronger influence in the intermediate water depth, while the choice of the hydrodynamic loading model has larger influence in deep water. Moreover, finite-depth FNV theory captures the loading in the widest range of wave and cylinder conditions. The areas of worst prediction by the numerical models were found to be the largest steepness and wave numbers for second harmonic, as well as the vicinity of the wave-breaking limit, especially for the third harmonic. The main cause is the non-monotonic growth of the experimental loading with increasing steepness due to flow separation, which leads to increasing numerical overpredictions since the numerical wave-loading models increase monotonically.

Nonlinear features of guided wave scattering from rivet hole nucleated fatigue cracks considering the rough contact surface condition

This article presents the investigation of nonlinear scattering features of guided waves from rivet hole nucleated fatigue cracks considering the rough contact surface condition. A small-size numerical model based on the local interaction simulation approach is developed, enabling the efficient analysis of the contact acoustic nonlinearity during the wave crack interactions. The study starts with an idealized breathing crack model possessing smooth, perfectly kissing contact surfaces. Then, the nature of rough crack surfaces is considered with randomly distributed initial openings and closures. Several distinctive aspects of the nonlinear scattering phenomenon are discussed: (1) the amplitude effect, which renders significantly different nonlinear response under various levels of excitation wave amplitudes; (2) the directivity and mode conversion features, which addresses the scattering direction dependence of the fundamental and superharmonic wave mode components; (3) the nonlinear resonance phenomenon, which maximizes the nonlinear response during the wave crack interactions at certain excitation frequency ranges. This study demonstrates that these nonlinear features are substantially influenced by the crack surface condition and differ much between an idealized breathing crack and a rough crack in most practical cases. Fatigue tests on a thin aluminum plate with a rivet hole is conducted to induce cracks in the specimen. An active sensor array surrounding the crack zone is implemented to generate and receive ultrasonic guided waves in various directions. Current work emphasizes on using a highly efficient numerical model to explain the nonlinear features of scattered waves from fatigue cracks considering the rough crack surface condition. These special features may provide insights and guidelines for nonlinear guided wave based nondestructive evaluation and structural health monitoring system design. The paper finishes with discussion, concluding remarks, and suggestions for future work.

Analyzing the stochastic behavior of ferroresonance initiation regarding initial conditions and system parameters

Power quality and power disturbances are the most important increasing factors throughout electrical networks. Ferroresonance as one of these disturbances can cause quality and security problems. This paper analyzes the stochastic behavior of ferroresonance as a nonlinear resonance phenomenon of the erratic nature and difficult prediction. Furthermore, the impact of different parameters on the initiation of ferroresonance is investigated. It is illustrated that a small change in the initial conditions leads to a large difference in the long-term behavior of the system, and this makes the future of the system unpredictable. Based on the stochastic behavior of ferroresonance some papers took different parameters into account and attempted to identify the basins of attraction for different ferroresonance regions. Here, according to real condition it is illustrated that these basins are highly depending on the initial conditions and there are no transparent boundaries for them. Also, by changing different parameters and initial conditions, the probability degree of ferroresonance initiation is calculated and depicted. Moreover, the impact of source angle on the ferroresonance initiation is investigated.

Multi-physics dynamics of a mechanical oscillator coupled to an electro-magnetic circuit

The dynamics of a non-linear electro-magneto-mechanical coupled system is addressed. The non-linear behavior arises from the involved coupling quadratic non-linearities and it is explored by relying on both analytical and numerical tools. When the linear frequency of the circuit is larger than that of the mechanical oscillator, the dynamics exhibits slow and fast time scales. Therefore the mechanical oscillator forced (actuated) via harmonic voltage excitation of the electric circuit is analyzed; when the forcing frequency is close to that of the mechanical oscillator, the long term damped dynamics evolves in a purely slow timescale with no interaction with the fast time scale. We show this by assuming the existence of a slow invariant manifold (SIM), computing it analytically, and verifying its existence via numerical experiments on both full- and reduced-order systems. In specific regions of the space of forcing parameters, the SIM is a complicated geometric object as it undergoes folding giving rise to hysteresis mechanisms which create a pronounced non-linear resonance phenomenon. Eventually, the roles played by the electro-magnetic and mechanical components in the resulting complex response, encompassing bifurcations as well as possible transitions from regular to chaotic motion, are highlighted by means of Poincaré sections.

Nonlinear resonance phenomenon of one-dimensional Bose—Einstein condensate under periodic modulation

We investigate the effect of an external periodic modulation on the one-dimensional (1D) Bose—Einstein condensate with harmonic trapping potential. By numerically solving the Gross—Pitaevskii equation with symplectic algorithm, the nonlinear resonance phenomenon is shown and the corresponding Fourier spectrum is given. The autoresonance phenomenon is also presented under almost periodic external modulation, and it shows that the condensate eventually evolves into quasi-periodic oscillation.