Weak Nonlinear Systems Research Articles

Identification of the form of self-excited aerodynamic force of bridge deck based on machine learning

This paper introduces an intelligent identification method for self-excited aerodynamic equations. The method is based on advanced sparse recognition technology and equipped with a new sampling strategy designed for weak nonlinear dynamic systems with limit cycle characteristics. Considering the complexity of the experiment condition and the difficult a priori selection of hyperparameters, a method based on information criteria and ensemble learning is proposed to derive the global optimal aerodynamic self-excited model. The proposed method is first validated by simulated data obtained from some well-known equations and then applied to the identification of flutter aerodynamic equations based on wind tunnel experiments. Finally, reasons for the different sparse recognition results under different sizes of candidate function space are discussed from the perspective of matrix linear correlation and numerical calculation.

Nonmonotonic dependence of thermal conductivity on surface roughness: A multiparticle Lorentz gas model.

Utilizing surface roughness to manipulate thermal transport has aided important developments in thermoelectrics and heat dissipation in microelectronics. In this paper, through a multiparticle Lorentz gas model, it is found that thermal conductivity oscillates with the increase of surface roughness, and the oscillating thermal conductivity gradually disappears with the increase of nonlinearity. The transmittance analyses reveal that the oscillating thermal conductivity is caused by localized particles due to boundary effects. Nonlinearity will gradually break the localization. Thus, localization still exists in the weak nonlinear system, where there exists an interplay between nonlinear interaction and localization. Furthermore, it is also found that boundary shapes have a great influence on the oscillating thermal conductivity. Finally, we have also studied the oscillating thermal rectification effects caused by rough boundaries. This study gains insight into the boundary effect on thermal transport and provides a mechanism to manipulate thermal conductivity.

Stochastic Resonance with Parameter Estimation for Enhancing Unknown Compound Fault Detection of Bearings.

Although stochastic resonance (SR) has been widely used to enhance weak fault signatures in machinery and has obtained remarkable achievements in engineering application, the parameter optimization of the existing SR-based methods requires the quantification indicators dependent on prior knowledge of the defects to be detected; for example, the widely used signal-to-noise ratio easily results in a false SR and decreases the detection performance of SR further. These indicators dependent on prior knowledge would not be suitable for real-world fault diagnosis of machinery where their structure parameters are unknown or are not able to be obtained. Therefore, it is necessary for us to design a type of SR method with parameter estimation, and such a method can estimate these parameters of SR adaptively by virtue of the signals to be processed or detected in place of the prior knowledge of the machinery. In this method, the triggered SR condition in second-order nonlinear systems and the synergic relationship among weak periodic signals, background noise and nonlinear systems can be considered to decide parameter estimation for enhancing unknown weak fault characteristics of machinery. Bearing fault experiments were performed to demonstrate the feasibility of the proposed method. The experimental results indicate that the proposed method is able to enhance weak fault characteristics and diagnose weak compound faults of bearings at an early stage without prior knowledge and any quantification indicators, and it presents the same detection performance as the SR methods based on prior knowledge. Furthermore, the proposed method is more simple and less time-consuming than other SR methods based on prior knowledge where a large number of parameters need to be optimized. Moreover, the proposed method is superior to the fast kurtogram method for early fault detection of bearings.

Wavenumber-space band clipping in nonlinear periodic structures

In weakly nonlinear systems, the main effect of cubic nonlinearity on wave propagation is an amplitude-dependent correction of the dispersion relation. This phenomenon can manifest either as a frequency shift or as a wavenumber shift depending on whether the excitation is prescribed as an initial condition or as a boundary condition, respectively. Several models have been proposed to capture the frequency shifts observed when the system is subjected to harmonic initial excitations. However, these models are not compatible with harmonic boundary excitations, which represent the conditions encountered in most practical applications. To overcome this limitation, we present a multiple scales framework to analytically capture the wavenumber shift experienced by dispersion relation of nonlinear monatomic chains under harmonic boundary excitations. We demonstrate that the wavenumber shifts result in an unusual dispersion correction effect, which we term wavenumber-space band clipping. We then extend the framework to locally resonant periodic structures to explore the implications of this phenomenon on bandgap tunability. We show that the tuning capability is available if the cubic nonlinearity is deployed in the internal springs supporting the resonators.

Detecting the Early Damages in Structures With Nonlinear Output Frequency Response Functions and the CNN-LSTM Model

Frames, shells, and hybrid structures with early damages, such as early cracks, often behave as extremely weak nonlinear systems, among which the nonlinearity is difficult to be detected, especially if the system response is affected by the noise. To avoid these damages becoming catastrophic failures, developing effective incipient damages detection methods is important. The nonlinear output frequency response functions (NOFRFs) and associated indexes can be considered as one kind of the prospective detection tools, which are usually determined from the established nonlinear autoregressive with exogenous inputs (NARX) model. However, the hyperparameters in the NARX model are difficult to be determined so that the identification accuracy cannot be guaranteed. Therefore, it is important to develop more accurate methods to estimate the NOFRFs and their associated indicators for damage detection. Motivated by the powerful learning abilities of convolutional neural networks (CNN) and long short-term memory (LSTM) networks, a novel method based on NOFRFs and the CNN-LSTM model for detecting the early damages in structures is proposed. By applying the beat excitation, the response of the structure is divided into two components, where the approximately linear component is used to estimate the frequency characteristic of the linear component by the classical linear model and the nonlinear component is used to establish the CNN-LSTM model. By calculating the responses of the two models, the NOFRFs and associated indexes can be accurately estimated, and then the early damage can be detected. Simulation and experimental studies verify the potential and effectiveness of the novel method proposed in this article.

Parameter matched stochastic resonance with damping for passive sonar detection

Stochastic resonance (SR) has been proven effective for weak signal detection under low signal-to-noise ratio (SNR) conditions. In this paper, a parameter matched stochastic resonance (PMSR) method is proposed to enhance the detection and extraction ability to weak signatures of moving vessels. Theoretical matched framework is established with a damped bistable SR model by optimizing the input-output signal-to-noise ratio improvement (SNRI) to nonlinear system parameters. By selecting a proper damping factor within a determinate constraint range, a weak periodic signal, background noise, and nonlinear system can be matched in generating a desired optimal output in the regime of matched parameter relationship. Then, we propose a PMSR based energy detection (ED) algorithm by taking the fully advantage of the Lorentzian characteristics. Numerical simulation analyses and application verifications are carried out to validate the effectiveness and efficiency of the proposed method, which reflects an excellent enhancement performance especially for low-frequency ship signatures.

Time‐Domain Substructure Transient Vibration Transfer Path Analysis Based on Time‐Varying Frequency Response Functions under Operational Excitations

The time‐domain substructure inverse matrix method has become a popular method to detect and diagnose problems regarding vehicle noise, vibration, and harshness, especially for those impulse excitations caused by roads. However, owning to its reliance on frequency response functions (FRFs), the approach is effective only for time‐invariable linear or weak nonlinear systems. This limitation prevents this method from being applied to a typical vehicle suspension substructure, which shows different nonlinear characteristics under different wheel transient loads. In this study, operational excitation was considered as a key factor and applied to calculate dynamic time‐varying FRFs to perform accurate time‐domain transient vibration transfer path analysis (TPA). The core idea of this novel method is to divide whole coupled substructural relationships into two parts: one involved time‐invariable components; normal FRFs could be obtained through tests directly. The other involved numerical computations of the time‐domain operational loads matrix and FRFs matrix in static conditions. This method focused on determining dynamic FRFs affected by operational loads, especially the severe transient ones; these loads are difficult to be considered in other classical TPA approaches, such as operational path analysis with exogenous inputs (OPAX) and operational transfer path analysis (OTPA). Experimental results showed that this new approach could overcome the limitations of the traditional time‐domain substructure TPA in terms of its strict requirements within time‐invariable systems. This is because in the new method, time‐varying FRFs were calculated and used, which could make the FRFs at the system level directly adapt to time‐varying systems from time to time. In summary, the modified method extends TPA objects studied in time‐invariable systems to time‐varying systems and, thus, makes a methodology and application innovation compared to traditional the time‐domain substructure TPA.

Classification of Hyperspectral Image by CNN Based on Shadow Area Enhancement Through Dynamic Stochastic Resonance

Target detection and classification in the shadow area of hyperspectral images (HSIs) has always been an important problem in the field of HSI data processing. However, there are few methods to detect or classify targets in the shadow area effectively because of occlusion of objects or oblique solar radiation. Dynamic stochastic resonance (DSR) theory shows that under the synergistic action of weak input signal, noise and non-linear system, the energy of noise can be transferred into signal partially. It breaks the idea that signal can be enhanced only by eliminating noise and has been proved to be effective in many fields. In this paper, DSR is introduced into the shadow area enhancement of HSI from both spatial and spectral dimensions. Then, the enhanced HSI data can be obtained by fusing the processed shadow with the original HSI data. Finally, 2D convolutional neural networks (2D-CNN) is used to classify the enhanced HSI. To evaluate the proposed method, a real-world HSI has been processed in the experiment and the results show that DSR can improve the contrast and the spectral intensity of HSI shadow area. Compared to other state-of-the-art methods, the classification accuracy is better at containing different targets with small samples, especially in the spectral dimension.

Displacement reduction effect and simplified evaluation method for SDOF systems using a clutching inerter damper

SummaryThis study evaluates the response reduction effect of linear single degree of freedom systems with a clutching inerter damper (CID) via parametric analysis under harmonic excitations and real earthquake records. The cause of the displacement reduction effect of a CID is inherited from the inertial mass damper (IMD)—reducing the nominal load intensity by increasing the mass by inertance. Additionally, the displacement reduction effect is further enhanced by the clutching effect, which speeds up the decreasing of the velocity response from an instantaneous extremum to 0. Thus, the CID is more effective than the IMD at reducing displacement responses. For example, the displacement response for a long‐period structure with a CID can be reduced by approximately 53%, while for an IMD, it can only be reduced by approximately 24%. Additionally, the linear single degree of freedom system with a CID is a weak nonlinear system reserving homogeneity, indicating that the response reduction factor will provide enough information to reveal the seismic reduction effect of the CID and that there is no need to consider the amplitude of the input excitations. To simplify the analysis of such nonlinear systems, an equivalent linearization method and a simplified formula of displacement reduction factors for code‐based designs are proposed and validated by another independent set of records from the European Strong‐motion Database.

Effects of Second-Order Matched Stochastic Resonance for Weak Signal Detection

Weak signal detection via stochastic resonance (SR) has attracted considerable attention in a wide range of research fields, especially under heavy background noise circumstances. In this paper, a second-order matched stochastic resonance (SMSR) method is proposed to further improve the signal-to-noise ratio of weak period signal. By selecting a proper damping factor in the regime of second-order parameter matched relationship, weak periodic signal, background noise, and nonlinear system can be matched in generating an enhanced output. The matched relationship is deduced in combining noise intensity optimization and signal frequency synchronization with duffing system in a mathematical way, and a normalized scale transformation is further carried out to make it accessible in detecting arbitrary high frequency signals. The numerical analysis and application verification are performed to confirm the validity and effectiveness of theoretical results, which indicate the proposed SMSR method is superior to the first-order parameter matched stochastic resonance in achieving a good band-pass filtering effect with a low-noise output as the driving frequency of received signal is not too small ( $\geqslant 0.1$ Hz). Thus, the proposed method is beneficial to practical engineering weak signal processing and anticipates to be a potential novel technique for ship radiated line-spectrum detection.

An attenuation signal-based identification approach for parameters in weak nonlinear systems with asymmetry

This paper introduces a method to identify the parameters in a weak nonlinear vibration system. Based on the existed backbone curve identification method which can identify the parameters in weak nonlinear symmetric systems, method in the present paper extends the application of ‘backbone curve’ method to nonlinear systems with asymmetry. Based on the introducing of offset term caused by the asymmetry, this method can increase the number of identification equations to broaden its application. Meanwhile, the data obtained from the displacement attenuation signal offer valuable information of system nonlinearity in this method. So we can identify all the parameters of system only from its displacement attenuation signal. As an application, we design a vertical cantilever beam with initial deflection and tip mass as an experimental object with asymmetric structure. The displacement data of the top of beam are acquired by a laser vibrometer. The provided method is applied to identify all nonlinear parameters in the model of the nonlinear cantilever beam. Then, the validity and accuracy of method can be evaluated by the comparison between theoretical parameters and identified parameters of beam model.

Control system design of flying-wing UAV based on nonlinear methodology

Abstract In this paper, A fluid vector rudder flying-wing UAV is employed as the design object, so as to study the nonlinear design method and flight validation. For the maneuvering flight control, this paper presents a control structure. This control structure included the inner loop linearization decoupling methods to eliminate the known negative coupling and the outer loop backstepping methods for trajectory tracking control. The stability of the control structure has been proved in this paper. Compared with the traditional backstepping control method, this controller increases the inner loop decoupling structure and retains the aerodynamic damping term which makes the linearized system a weak nonlinear system. This structure can not only reduce the conservatism of the outer loop controller design, but also is convenient for engineering implementation. Simulation and flight validation results show that the proposed control scheme is effective.

Dynamical phonon laser in coupled active-passive microresonators

Effective transition between the population-inverted optical eigenmodes of two coupled microcavities carrying mechanical oscillation realizes a phonon analogue of optical two-level laser. By providing an approach that linearizes the dynamical equations of weak nonlinear systems without relying on their steady states, we study such phonon laser action as a realistic dynamical process, which exhibits time-dependent stimulated phonon field amplification especially when one of the cavities is added with optical gain medium. The approach we present explicitly gives the conditions for the optimum phonon lasing, and thermal noise is found to be capable of facilitating the phonon laser action significantly.

Higher eigenmodes of matter wave solitons in parity-time symmetric complex potential

Summary This work analyzes the higher eigenmodes of matter wave solitons in parity-time (PT) symmetric complex potential using variational and numerical methods. The dynamical states of the system have been investigated both in the linear and the nonlinear regime. In the linear case the particle number (power in optics) of the higher eigen mode remains constant for low values of imaginary part of the potential (gain/loss term). The system exhibits a spontaneous phase transition from PT symmetric regime to broken PT symmetric regime. The PT phase transition point depends on the strength of the imaginary component of the potential. The system has also been studied by applying the self-focusing cubic nonlinearity. For low values of the strength of the imaginary part of the potential the stable matter wave solitons are formed. The PT symmetric regime strongly depends on the strength of the nonlinear coefficient and the chemical potential (the propagation constant). For large values of the nonlinearity the particle number of the higher eigenmode increases which leads to the collapse of the solitons. The stable solitons are formed in a weak nonlinear system with low values of the strength of the imaginary potential. The variational results are found to be in close agreement with numerical results.

On the Interval Stability of Weak-Nonlinear Control Systems with Aftereffect.

Sufficient conditions of interval absolute stability of nonlinear control systems described in terms of systems of the ordinary differential equations with delay argument and also neutral type are obtained. The Lyapunov-Krasovskii functional method in the form of the sum of a quadratic component and integrals from nonlinearity is used at construction of statements.

Strong and weak nonlinear dynamics: Models, classification, examples

The difference between strong and weak nonlinear systems is discussed. A classification of strong nonlinearities is given. It is based on the divergence or inanity of series expansions of the equation of state commonly used in the study of weak nonlinear phenomena. Such power or functional series cannot be used in three cases: (i) if the equation of state contains a singularity; (ii) if the series diverges for strong disturbances; (iii) if the linear term is absent, and higher nonlinearity dominates. Strong nonlinearities are known in acoustics, optics, mechanics and in quantum field theory. Mathematical models, solutions and observed phenomena are presented. For example, an equation of Heisenberg type and its generalization for strongly nonlinear wave system are given. In particular, exact solutions of new “quadratically cubic” Burgers and Riemann-Hopf equations are discovered.

Assessment of Perturbation and Homotopy-Perturbation Methods for Solving Nonlinear Oscillator Equations

In this paper, two modified perturbation methods, namely, artificial parameter method (APM) and homotopy perturbation method (HPM) have been successfully implemented to find the solution of van der Pol nonlinear oscillator equation. Different from classical perturbation method, APM and HPM do not require small parameter and therefore, obtained approximate solutions may be uniformly valid for both weak nonlinear systems and strong nonlinear systems. Comparison of the results obtained by the proposed methods reveals that APM and HPM are more effective compared to classical perturbation method and with only a few terms, approximate the exact solution with a fairly reasonable error.

Study on the Calculation Methods for Weak-Nonlinear Rotor System and its Response

The operation condition of the rotor system is strong nonlinear state; even with severe breakdown, weak nonlinear state appears; due to the nature of the rotor system is nonlinear. The system of the weak nonlinear state could mutate for strong nonlinear state. So the rotor system of weak nonlinear state vector in the system analysis and fault diagnosis is very important. Therefore, this paper analyzes universality of weak nonlinear of the system. It introduces segmentation system displacement vector into two little freedom of time to approximate equation system vector into a order system method.

Determination of applicable input range for approximating a nonlinear FGR furnace around the design point

In this paper, the nonlinear dynamic characteristics of a FGR furnace have been analysed around the furnace design point. Based on the steady-state results of full-scale nonlinear CFD simulations, the maximal allowable range on the variations of the furnace inputs can be determined, once for the maximal error bound between nonlinear system and its linear counterpart is specified. It is interesting to note that for a reheating furnace, the nonlinearities associated with the heat load are less severe than that associated with NO emission. With due consideration of the established input signal linear ranges, the linearized dynamic models of the furnace are derived by applying system identification technologies using the data generated from the CFD simulations. Analysis and validation of the models are also carried out. It is concluded that this technique is applicable to weak nonlinear systems around the design point. The results of the analysis provide additional insights on the nature of the nonlinearities as well as guidelines for selecting the input amplitude if system identification techniques are used. So long as the amplitudes of the probing signals satisfy the respective input constraints, the obtained linearized models will be applicable around the design point. Subsequently, these models can be used to design feedback controllers to maintain the furnace operated around the design point.

INTERNAL RESONANCES AND BIFURCATIONS OF AN ARRAY BELOW THE FIRST PULL-IN INSTABILITY

A nonlinear continuum model is used to investigate the dynamic behavior of an array of N nonlinearly coupled microbeams. Investigations concentrate on the region below the array's first pull-in instability, which is shown to be governed by several internal three-to-one and combination resonances. The nonlinear equations of motion for a two-element system are solved using the asymptotic multiple-scales method for the weak nonlinear system. The analytically obtained periodic response of two coupled microbeams is numerically evaluated by a continuation technique and complemented by a numerical analysis of a three-element array which exhibits quasi-periodic responses and lengthy chaotic transients. This study of small-size microbeam arrays serves for design purposes and the understanding of nonlinear nearest-neighbor interactions of medium- and large-size arrays.