Piecewise Nonlinear System Research Articles

The identification of piecewise non-linear dynamical system without understanding the mechanism.

This paper designs an algorithm to distill the piecewise non-linear dynamical system from the data without prior knowledge. The system to be identified does not have to be written as a known model term or be thoroughly understood. We exploit the fact that an unknown piecewise non-linear system can be decomposed into the Fourier series as long as its equations of motion are Riemann integrable. Based on this property, we reduce the challenge of finding the correct model to discovering the Fourier series approximation. However, the Fourier series approximation of the piecewise function is inaccurate. The new method takes advantage of this weakness to determine whether the model has piecewise features and to find a way to discover the discontinuity set. Then, the dynamical system on each segment is identified as a pure Fourier series. Identification of intricate models can be achieved in simple steps. The results show that the method can accurately discover the equation of motion and precisely capture the non-smooth characteristic. Next, the prediction and further detailed analysis can be carried out.

Study on the dynamics of two-degree-of-freedom fractional differential piecewise nonlinear systems under harmonic excitation

In this paper, a two-degree-of-freedom piecewise nonlinear system model with fractional differential term is constructed. The incremental harmonic balance method (IHB) is used to solve the two-degree-of-freedom piecewise nonlinear system with fractional differential term and the amplitude–frequency curve of the system is obtained. The correctness of the solution of the incremental harmonic balance method is verified by numerical methods. Then the influence of system parameters on the amplitude–frequency response characteristics of the system is studied. The bifurcation diagram of the system is analyzed by numerical method, and the dynamic characteristics of the system under typical parameters are obtained.

Principal resonance analysis of piecewise nonlinear oscillator with fractional calculus

To analyze the piecewise non-linear system with fractional order differentiation, fractional-order was introduced into the two piecewise systems to accurately describe the stress relaxation of viscoelastic materials. A non-linear dynamic model with non-linear stiffness, damping, and fractional-order multipiece wise points was also established. Under periodic excitation, the equation of the non-linear system relationship in the system was obtained using the average method, where the amplitude-frequency response characteristics under different damping, stiffness, and fractional order parameters were provided. The influence of non-linear factors and fractional order terms on the stability of the system was studied. The chaotic behavior of the system under different parameter disturbances was determined, and the results indicate that the system presents chaotic behavior with the change in disturbance parameters. Moreover, the decreased linear damping subjects the system to a wider range of chaotic states.

Study on a Class of Piecewise Nonlinear Systems with Fractional Delay

In this paper, a dynamic model of piecewise nonlinear system with fractional-order time delay is simplified. The amplitude frequency response equation of the dynamic model of piecewise nonlinear system with fractional-order time delay under periodic excitation is obtained by using the average method. It is found that the amplitude of the system changes when the external excitation frequency changes. At the same time, the amplitude frequency response characteristics of the system under different time delay parameters, different fractional-order parameters, and coefficient are studied. By analyzing the amplitude frequency response characteristics, the influence of time delay and fractional-order parameters on the stability of the system is analyzed in this paper, and the bifurcation equations of the system are studied by using the theory of continuity. The transition sets under different piecewise states and the constrained bifurcation behaviors under the corresponding unfolding parameters are obtained. The variation of the bifurcation topology of the system with the change of system parameters is given.

A new method for weak fault feature extraction based on piecewise mixed stochastic resonance

In a continuous bistable system, when the input signal is continuously increased, the output signal tends to be stable and no longer increases. At this time, the weak signal under strong background noise is difficult to be extracted, which means saturation occurs. Aiming at the saturation characteristics of stochastic resonance (SR), the proposed piecewise nonlinear bistable system (PNBSR) model has achieved certain results. However, the potential barrier in the middle of the PNBSR method still completely uses the potential function of classical bistable stochastic resonance (CBSR). There is no fundamental solution to the fourth-order limitation. This paper explores an improved piecewise mixed stochastic resonance (PMSR) potential model. The fourth-order potential function that restricts particle motion in CBSR is improved to a piecewise second-order potential function. This potential function subverts the shape of the traditional potential function and presents a symmetrical double-hook shape. Based on PMSR model, this paper uses particle swarm optimization (PSO) to select system parameters and elaborates the characteristics of the potential function curve in detail. Under the same conditions, the output signal-to-noise ratio (SNR) curve of the improved system is generally higher than that of the CBSR and PNBSR systems. Experiments on bearings and gears show that the proposed method can accurately extract weak fault features, and the effect is better than the PNBSR method.

A device capable of customizing nonlinear forces for vibration energy harvesting, vibration isolation, and nonlinear energy sink

Nonlinear forces obtained via springs, magnets, buckling beams, etc., have been widely introduced into vibration energy harvesting, vibration isolation and nonlinear energy sink to improve performance. Although obtaining customized nonlinear forces as desired is difficult, it is quite useful to improve the dynamic performance. Therefore, a device capable of customizing nonlinear forces is developed in this study. The device consists of a pre-compressed spring, a bearing and a raceway, which are designed according to the nonlinear forces to be customized. It is assumed that the nonlinear forces to be customized are continuous (not necessarily smooth). As examples, a hardening nonlinear system, a softening nonlinear system, a bistable system, a tri-stable system, and a piecewise nonlinear system are customized in this study. Simulations and experiments prove its feasibility. The device enables designers to obtain the most suitable nonlinear forces for the dynamic systems. It can be easily applied to energy harvesting, vibration isolation, and nonlinear energy sink, etc.

Statistical Linearization Method for the Piecewise Nonlinear Systems

The Monte-Carlo numerical simulation method is applied to obtain the approximate probability distributions of the response of the piecewise nonlinear systems to random excitation. The expression of the equivalent linearization parameters for the piecewise nonlinear system is derived by the proposed statistical linearization method. To illustrate the accuracy of the linearization method, the present paper has compared the statistical characteristic value obtained by both the Monte Carlo numerical simulation method and the Statistical linearization method, while the time history analysis also shows a good agreement. Linearization of piecewise stiffness nonlinear model is displayed.

Stochastic Resonance in Unsaturated Piecewise Nonlinear Bistable System Under Multiplicative and Additive Noise for Bearing Fault Diagnosis

With regard to the fault diagnosis, the stochastic resonance (SR) method takes advantage of noise imbedded in vibration signals while most traditional methods suppress or eliminate noise to enhance weak fault characteristics. In this paper, a novel piecewise nonlinear bistable SR (PNBSR) is proposed and its corresponding potential function called piecewise nonlinear bistable system (PNBS) overcomes the output saturation disadvantage which bothers the classical bistable system (CBS). The output saturation limits the enhancement capability for extracting fault characteristics of classical bistable SR (CBSR). Satisfying the adiabatic condition, the expression of the output signal-to-noise (SNR) of PNBSR is derived and compared with the output SNR of CBSR. Considered the multiplicative and additive noise, two methods are applied to extract characteristic frequency from simulated harmonic vibration signal and actual bearing fault signals. The SNR increase (SNRI) is chosen as the index for evaluating the performance of CBSR and PNBSR in experimental simulations. The diagnosis results indicate that the proposed PNBSR method is superior to the CBSR by analyzing SNRI and the effect of extracting fault characteristics.

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.

Bifurcation and chaos for piecewise nonlinear roll system of rolling mill

A non-smooth cold roll system of rolling mill is studied to reveal the bifurcation of the piecewise-smooth and discontinuous system. To examine the influence of the parameters on the dynamics, the bifurcation diagram is constructed when it is unperturbed. Hamilton phase diagrams of the non-smooth system are detected, which differ significantly from the ones obtained in the smooth system. Non-smooth homoclinic, heteroclinic, and periodic orbits are determined, which depend on the classical heteroclinic periodic orbits, periodic orbits, and a necessary condition. A extended Melnikov function is employed to obtain the criteria for the non-smooth homoclinic bifurcation in this class of piecewise-smooth and discontinuous system, which implies that the existence of homoclinic bifurcation arises from the breaking of homoclinic orbits under the perturbation of damping. The results reveal that this kind of non-smooth factor has little influence on the chaotic threshold apart from calculating piecewise integrals. T...

Research on Multi-signal Based Neuro-fuzzy Hammerstein-Wiener Model

In order to solve the control problem of complex systems,it is important to design a special structure model with data information to simplify the question of designing control system.Thus,a multi-signal based neuro-fuzzy HammersteinWiener model is proposed,which breaks through the traditional iterative separation method.The separation of the neuro-fuzzy nonlinear and linear parts of the Hammerstein-Wiener model is realized by one kind of multi-signals.And a noniterative neuro-fuzzy optimization algorithm is designed to expand the research results to piecewise nonlinear system,which can be applied to much more nonlinear systems.This algorithm guarantees the precision of the model.Moreover,it has the ability of approximating strong nonlinearity.Furthermore,a neuro-fuzzy Hammerstein-Wiener model based control system is designed to simplify the control problem of the nonlinear system into the problem of linear system by using the special structure of the model.As a result,the traditional PID controller can get a better control result.Simulated results show the effectiveness of the method.

Stochastic resonance in a piecewise nonlinear system driven by colored correlated additive and multiplicative colored noises

In this paper, we study the stochastic resonance in a piecewise nonlinear system driven by a periodic signal and colored noises, which is described by multiplicative and additive colored noises with colored cross-correlation. Using the two-state theory and the unified colored approximation, we can derive the analytical expressions of the steady-state probability density and the signal-to-noise ratio (SNR). Effects of colored noises and the periodic signal on SNR are presented. It is found that the conventional stochastic resonance and bona-fide stochastic resonance may exist in this system. Moreover, the value of the SNR peak decreases with increasing correlation time and correlation between the additive and multiplicative noises.

The mean first-passage time for piecewise nonlinear system driven by colored correlated additive and multiplicative colored noises

In this paper, We have studied the effects of intensity and correlation time of noises on the mean first-passage time in a picecewise nonlinear system driven by multiplicative and additive colored noises with colored cross-correlation. We derived the expression of the mean first-passage time (MFPT) by applying the unified colored approximation method and the steepest-descent approximation. Results show that the MFPT of the system exhibits a mono-peak structure and the “resonance” phenomena enhance with the increase of multiplicative noise intensity. The value of the peak decreases with increasing additive noise intensity and the correlation between the additive and multiplicative noises. However, the MFPT of the system increases with the increase of additive noise intensity. That is, the effects of the additive noise and the multiplicative noise on MFPT are different. Moreover, the negative and passive correlations play different roles in the MFPT.

The global responses characteristics of a rotor/stator rubbing system with dry friction effects

Rotor/stator rubbing systems may undertake a number of quite different responses. Recent experiments on rotor/stator rubbing have revealed that two or three different responses may coexist. In this paper the global response characteristics of a general rotor/stator rubbing system, which takes into account the dominant factors in the process of rotor/stator rubbing, especially, the dry friction effect that is mostly neglected in the previous works and is the main factor for the self-excited dry friction backward whirl, are studied. The different solutions of the piecewise nonlinear system are derived and their stability are analyzed to get the existence boundaries of the different responses. An overall picture of the global response characteristics of this model is then obtained by drawing the existence boundaries in a same parameter space. The present results provide good understanding on the coexistence of different rubbing responses observed in tests. Moreover, deeper insight into the types of coexistence of different rubbing responses and their relationship with the system parameters is gained.

Suppressing grazing chaos in impacting system by structural nonlinearity

In this note we investigate the influence of structural nonlinearity of a simple cantilever beam impacting system on its dynamic responses close to grazing incidence by a means of numerical simulation. To obtain a clear picture of this effect we considered two systems exhibiting impacting motion, where the primary stiffness is either linear (piecewise linear system) or nonlinear (piecewise nonlinear system). Two systems were studied by constructing bifurcation diagrams, basins of attractions, Lyapunov exponents and parameter plots. In our analysis we focused on the grazing transitions from no impact to impact motion. We observed that the dynamic responses of these two similar systems are qualitatively different around the grazing transitions. For the piecewise linear system, we identified on the parameter space a considerable region with chaotic behaviour, while for the piecewise nonlinear system we found just periodic attractors. We postulate that the structural nonlinearity of the cantilever impacting beam suppresses chaos near grazing.

Period-one motions of a mechanical oscillator with periodically time-varying, piecewise-nonlinear stiffness

The dynamic behavior of a piecewise-nonlinear mechanical oscillator with parametric and external excitations is investigated. The viscously damped single-degree-of-freedom oscillator is subjected to a periodically time-varying, piecewise-nonlinear restoring function defined by a clearance surrounded by continuously nonlinear (quadratic and cubic) regions. Typical applications represented by this oscillator are highlighted. A multi-term harmonic balance formulation is used in conjunction with discrete Fourier transforms and a parametric continuation scheme to determine steady-state period-1 motions of the system due to both parametric and external excitations. The accuracy of the analytical solutions is demonstrated by comparing them to direct numerical integration solutions. Floquet theory is applied to determine the stability of the steady-state harmonic balance solutions. At the end, detailed parametric studies are presented to quantify the combined influence of clearance, quadratic and cubic nonlinearities within reasonable ranges of all other system parameters. A comparison between time-varying and time-invariant systems is also provided to demonstrate the influence of the parametric and external excitations on a piecewise-nonlinear system.

Dynamics of a piecewise non-linear system subject to dual harmonic excitation using parametric continuation

The dynamic behavior of piecewise linear and/or non-linear vibratory systems subject to a single harmonic excitation has been recently analyzed by using a one- or multi-term harmonic balance (or Galerkin’s) method, piecewise linear techniques, analog simulation and direct numerical integration (digital simulation). Instead, this paper proposes to utilize the technique of parametric continuationto study the steady state response and global dynamics of a two-degree-of-freedom piecewise non-linear system with backlash or multi-valued springs and impact damping. The physical system is under the influence of a mean load and is subject to a dual harmonic external excitation, where the second frequency is either twice or three times the fundamental excitation frequency. The effects of various parameters such as impact damping, mean and alternating loads, the piecewise stiffness ratios and the relative phase between the excitation harmonics on the system response are thoroughly examined. System or excitation parameter regimes that exhibit sub-harmonic, super-harmonic quasi-periodic and chaotic solutions are obtained efficiently and systematically by using the proposed scheme. This investigation also clarifies the differences between viscous and non-linear impact damping models, which have been proposed earlier in the literature for studying clearance non-linearities. Limited experimental data validate our modelling strategy. Finally, selected predictions from this technique match very well with a multi-term harmonic balance method and with direct numerical integration, wherever applicable.

A Toeplitz Jacobian Matrix/Fast Fourier Transformation Method for Steady-State Analysis of Discontinuous Oscillators

A semianalytical algorithm is proposed for the solutions and their stability of a piecewise nonlinear system. The conventional harmonic balance method is modified by the introduction of Toeplitz Jacobian matrices (TJM) and by the alternative applications of fast Fourier transformation (FFT) and its inverse. The TJM/FFT method substantially reduces the amount of computation and circumvents the necessary numerical differentiation for the Jacobian. An arc-length algorithm and a branch switching procedure are incorporated so that the secondary branches can be independently traced. Oscillators with piecewise nonlinear characteristics are taken as illustrative examples. Flip, fold, and Hopf bifurcations are of interest.

A Toeplitz Jacobian Matrix/Fast Fourier Transformation Method for Steady-State Analysis of Discontinuous Oscillators

A semianalytical algorithm is proposed for the solutions and their stability of a piecewise nonlinear system. The conventional harmonic balance method is modified by the introduction of Toeplitz Jacobian matrices (TJM) and by the alternative applications of fast Fourier transformation (FFT) and its inverse. The TJM/FFT method substantially reduces the amount of computation and circumvents the necessary numerical differentiation for the Jacobian. An arc-length algorithm and a branch switching procedure are incorporated so that the secondary branches can be independently traced. Oscillators with piecewise nonlinear characteristics are taken as illustrative examples. Flip, fold, and Hopf bifurcations are of interest.