Tristable System Research Articles

Stochastic Resonance Effect in Segmented Underdamped Asymmetric Tristable System and Its Application in Weak Signal Detection Research

Stochastic Resonance Effect in Segmented Underdamped Asymmetric Tristable System and Its Application in Weak Signal Detection Research

Coupled piecewise tri-stable stochastic resonance system driven by dual inputs and joint EMD/VMD analysis

A novel piecewise tri-stable stochastic resonance (NPTSR) system is proposed to address the issue of output saturation caused by high-order terms limitation in this paper. Building upon this, the exploration is extended to a coupled piecewise tri-stable stochastic resonance system driven by dual inputs (DCPTSR). First, we uncover the influence of dual input interaction on output quality, finding that when the fRequencies of the two input signals are consistent, varying the amplitude of the driving signal can effectively enhance the output performance of the target signal. Secondly, by utilizing the adiabatic approximation theory, the steady-state probability density (SPD) and signal-to-noise ratio (SNR) of the DCPTSR system are derived, which allows us to analyze the effects of various parameters on both SPD and SNR. Next, three combined denoising systems, namely EMD-DCPTSR, VMD-DCPTSR, and SDCPTSR, are constructed by utilizing empirical mode decomposition (EMD), variational mode decomposition (VMD), and the stochastic resonance (SR). Through numerical simulations, we demonstrate that the combined denoising system outperforms the stand-alone SR system, and we analyze the stochastic resonance phenomenon of the DCPTSR system using the spectral amplification (SA) coefficient as an evaluation index. Finally, to assess practical applicability, these systems are deployed for bearing fault detection. The experimental results exhibit notable signal-to-noise gain improvements for the DCPTSR system compared to standalone SR systems by 0.7699 ∼ 9.4541 dB. The EMD-DCPTSR system shows signal-to-noise gain improvements of 0.3245 ∼ 1.1709 dB compared to the VMD-DCPTSR and SDCPTSR systems. Moreover, all three combined denoising systems outperform the standalone SR system in terms of signal processing capabilities. In conclusion, this paper extensively investigates the interaction between the two input signals in a dual-input system and studies the output performance of using EMD, VMD, and SR as preprocessing methods for the SR system. Through numerical simulations and practical engineering applications, we highlight the substantial advantages of combined denoising systems. These findings offer essential theoretical insights and promising prospects for engineering applications.

Erratum to: Analysis of Weak Signal Detection Based on Tri-Stable System under Poisson White Noise

Erratum to: Analysis of Weak Signal Detection Based on Tri-Stable System under Poisson White Noise

Delay-independent stability of a tri-stable energy harvesting system with time-delayed feedback control

This paper analyzes the delay-independent stability and harvesting performance of a tri-stable vibrational energy harvester (TVEH) with time-delayed feedback control. The dynamical model of the controlled electromechanical TVEH is established, and the method of multiple scales is employed to derive the steady-state response near the primary resonance. The delay-independent stability conditions of the TVEH are obtained using Routh Hurwitz criterion and classical Sturm criterion. The influences of time-delayed feedback control on the TVEH's output and jump phenomena are discussed through the steady-state response. Furthermore, appropriate choices of feedback gains and time delays can ensure the system stability and enhance the output power. The theoretical results are validated through numerical simulations, demonstrating the effectiveness of the proposed control strategy.

Research and Application of Two-Dimensional Time-Delayed Tri-Stable Stochastic Resonance System for Bearing Fault Detection

The time-delayed feedback term can improve the output of a system, while the two-dimensional stochastic resonance (SR) system has a stronger signal amplification capability. To improve the output signal-to-noise ratio (SNR) of the system, this paper proposes a two-dimensional time-delayed tri-stable stochastic resonance system (TDTDTSR) based on the advantages of the above two systems. First, the steady-state probability density function (SPD), the mean first-pass time (MFPT), and the output SNR are derived under adiabatic approximation theory, and the effects of different system parameters on them are investigated. Next, TDTDTSR and the classical two-dimensional tri-stable stochastic resonance system (CTDTSR) system are simulated numerically. The results show that the mean signal-to-noise gain (MSNRG) of TDTDTSR system is higher than that of the CTDTSR system. Finally, the system parameters are optimized using a genetic algorithm, and the application of TDTDTSR to bearing fault detection is compared with CTDTSR and the novel piecewise symmetric two-dimensional tri-stable stochastic resonance (NPSTDTSR) systems. The experimental results demonstrate that TDTDTSR system has better performance, providing valuable theoretical support and practical engineering applications for the system in subsequent analyses.

Delay Segmented Tristable Stochastic Resonance System Driven by Non-Gaussian Colored Noise and Its Application in Bearing Fault Detection

Abstract This study investigates the Delayed Segmented Tristable Stochastic Resonance (DSTSR) system under the influence of additive non-Gaussian colored noise. The research employs an improved segmented tristable potential function, wherein the equilibrium points and barrier heights can be independently controlled by parameters. Simultaneously, the segmented function on both sides reduces the restrictions of higher-order terms on the walls of the potential wells. The equivalent Langevin equation for the DSTSR system is obtained using the path integral method, the unified colored noise approximation method, and the small-delay approximation. Subsequently, the theoretical expressions for the steady-state probability density, mean first passage time (MFPT), and Signal-to-Noise Ratio (SNR) are derived from the resulting equations, and the impact of variations in system parameters on these performance metrics is discussed. Additionally, Monte Carlo simulations for MFPT are conducted to verify the accuracy of the theoretical derivations. Combining the results from the theoretical section and the impact of parameters on system performance, the article employs an adaptive genetic algorithm to optimize system parameters. This algorithm is then applied to simulation experiments and bearing fault detection. In the simulation experiments, the DSTSR system is compared with other systems. The results indicate that the DSTSR system exhibits the highest SNR improvement. Furthermore, in bearing fault detection under non-Gaussian colored noise, the DSTSR system shows higher spectral amplitude and SNR at the fault frequency compared to the tristable stochastic resonance system and the segmented tristable stochastic resonance system without time delay feedback. This suggests that stochastic resonance can effectively detect weak signals in non-Gaussian non-white noise scenarios, and the introduction of time delay contributes to the occurrence of stochastic resonance to a certain extent.

An improved coupled tri-stable energy harvesting system with spring stops for passive control

Motivated by enhancing the DC power output of a tri-stable energy harvester (TEH), a passive control of a coupled TEH driven by colored noise is constructed by using spring stops. The cubic trilinear spring model is utilized to describe the piecewise nonlinear impact force introduced by the spring stops. Using the stochastic averaging technique based on energy-dependent frequency, the analytical expressions of joint probability density function (PDF) and harvested power are obtained. The effects of the impact distance, impact stiffness, and colored noise on stochastic dynamics and harvesting performance are further discussed. It is found that the spring stops can change the shape of the PDF to induce stochastic bifurcation phenomena. The TEH under passive control tends to oscillate among three potential wells to effectively improve the DC output. By adjusting the appropriate impact distance and stiffness, the harvesting performance can be optimized and improved. Moreover, the numerical simulations and experimental verifications well support the theoretical analysis. Based on the experimental results, the harvested power can be enhanced by choosing the appropriate material of the stop (i.e., the spring is the optimal one among three different materials).

Dynamics and vibration reduction performance of asymmetric tristable nonlinear energy sink

With its complex nonlinear dynamic behavior, the tristable system has shown excellent performance in areas such as energy harvesting and vibration suppression, and has attracted a lot of attention. In this paper, an asymmetric tristable design is proposed to improve the vibration suppression efficiency of nonlinear energy sinks (NESs) for the first time. The proposed asymmetric tristable NES (ATNES) is composed of a pair of oblique springs and a vertical spring. Then, the three stable states, symmetric and asymmetric, can be achieved by the adjustment of the distance and stiffness asymmetry of the oblique springs. The governing equations of a linear oscillator (LO) coupled with the ATNES are derived. The approximate analytical solution to the coupled system is obtained by the harmonic balance method (HBM) and verified numerically. The vibration suppression efficiency of three types of ATNES is compared. The results show that the asymmetric design can improve the efficiency of vibration reduction through comparing the chaotic motion of the NES oscillator between asymmetric steady states. In addition, compared with the symmetrical tristable NES (TNES), the ATNES can effectively control smaller structural vibrations. In other words, the ATNES can effectively solve the threshold problem of TNES failure to weak excitation. Therefore, this paper reveals the vibration reduction mechanism of the ATNES, and provides a pathway to expand the effective excitation amplitude range of the NES.

Novel compound multistable stochastic resonance weak signal detection

Abstract The research on stochastic resonance (SR) which is used to extract weak signals from noisy backgrounds is of great theoretical significance and promising application. To address the shortcomings of the classical tristable SR model, this article proposes a novel compound multistable stochastic resonance (NCMSR) model by combining the Woods–Saxon (WS) and tristable models. The influence of the parameters of the NCMSR systems on the output response performance is studied under different α stable noises. Meanwhile, the adaptive synchronization optimization algorithm based on the proposed model is employed to achieve periodic and non-periodic signal identifications in α stable noise environments. The results show that the proposed system model outperforms the tristable system in terms of detection performance. Finally, the NCMSR model is applied to 2D image processing, which achieves great noise reduction and image recovery effects.

Bifurcation analysis of a tristable system with fractional derivative under colored noise excitation

Stochastic bifurcation has received much attention recently and is of great significance in the research on the dynamics of nonlinear systems. In this paper, we study the bifurcation of a self-sustained tristable system containing fractional derivative under the excitation of two colored noises, where the self-sustained tristable system consists of stable limit cycles and a stable state. The multiple scale method and generalized harmonic function are used to convert the original system into an equal system without obvious time delay and fractional derivative. Then the stationary probability density function (SPDF) of the system is obtained by using the stochastic averaging method to discuss stochastic bifurcation. Based on singularity theory, the equation satisfying amplitude is obtained to derive the corresponding bifurcation diagram. From the bifurcation analysis of the system, it is revealed that fractional order, fractional coefficient, and the intensity and correlation time of colored noise can be utilized as bifurcation parameters, causing peculiar stochastic bifurcation phenomena and regulating the output of the system, as well as the relatively big colored noise intensity and relatively small correlation time facilitate the realization of large magnitude limit cycle. Monte Carlo numerical results verify the validity of the theoretical approach. In addition, appropriately changing the delayed feedback parameter helps to govern the bifurcation range and causes richer bifurcation phenomena. The results of this paper contribute to a better understanding of self-sustained tristable systems with fractional derivative and may provide reference value in resolving the problems of chattering in high-speed aircraft and applications to viscoelastic materials.

Parametric study and multi-parameter optimization of a generalized second-order tri-stable stochastic resonance system

Parametric study and multi-parameter optimization of a generalized second-order tri-stable stochastic resonance system

Multi-dimensional hybrid potential stochastic resonance and application of bearing fault diagnosis

Stochastic resonance is a method to enhance weak signal by using noise, which has a wide range of applications in weak signal detection. In order to further investigate the application of multi-dimensional coupling stochastic resonance in practical engineering, a Multi-dimensional Double connection coupling Hybrid Potential Stochastic Resonance (MDHPSR) system is proposed in this paper. Firstly, the potential functions of bi-stable and tri-stable are briefly described, and the tri-stable system having a higher output amplitude. Secondly, the effect of different coupling methods on the output of the central and adjacent coupling ends are studied, and the output amplitude of double connection coupling is higher. Then, the double connection coupling of different potential functions is studied, and MDHPSR system effect is the best. Compared with Multi-dimensional Double connection Classical Bi-stable Stochastic Resonance (MDCBSR) system, MDHPSR system has better anti-noise performance. Finally, applying the two multi-dimensional coupling systems to bearing fault diagnosis, MDHPSR system output amplitude is higher and the Signal-to-Noise Ratio (SNR) is improved by more than 3 dB. This demonstrates the superior performance of MDHPSR system for weak signal detection and the value of the multi-dimensional coupling system for practical engineering applications.

A cascaded piecewise unsaturated asymmetric under-damped tri-stable stochastic resonance system and its application in bearing fault diagnosis

A cascaded piecewise unsaturated asymmetric under-damped tri-stable stochastic resonance system and its application in bearing fault diagnosis

Numerical study of wake induced vibration(WIV) and energy harvesting characteristics under different stable conditions

In order to better capture energy from natural water flow, we designed a novel captive energy model using wake induced vibration (WIV) combined with multi-stable theory. The model consists of an upstream stationary square cylinder and a downstream vibrating circular cylinder. The multi-stable characteristics are achieved by controlling the horizontal distance (l) and vertical distance (d) between the tip magnet of the vibrating cylinder and the stationary magnets. We simulate the capture energy characteristics of the harvester for different stable conditions at 0.1 m/s-0.8 m/s (i.e., 1990 ≤ Reynolds number ≤ 15,923) using the computational fluid dynamics (CFD) method. The simulation results show that when fixing l = 0.020, d = 0.016 and d = 0.017 are bi-stable systems, and at the same time, it will be trapped in a potential well at a flow velocity of 0.1 m/s, resulting in a lower amplitude and vibration frequency. When d is fixed at 0.017 m, l = 0.021 and l = 0.022 are tri-stable systems with a wider flow velocity interval corresponding to its high amplitude. Different magnet positions affect the phase between the lift coefficients and displacements of the bluff body. At flow velocities from 0.1 m/s to 0.4 m/s, the energy harvesting effect of the with-magnet system is better than that of the non-magnet system.

4FSK signal processing based on adaptive tristable stochastic resonance under levy noise

Aiming at the problem of low quality of coherent demodulation transmission signal of digital signal in the background of Levy noise, this paper proposes a new method of 4FSK signal reception based on adaptive tristable stochastic resonance (SR) system. The bit error rate (BER) of output signals of adaptive tristable SR system model and the traditional modulation model are compared by simulation. The numerical simulation results show that compared with the traditional model, the output BER of the 4FSK signal after demodulation of the adaptive tristable SR system model is reduced by 4% compared with the traditional model. The time domain image of 4FSK signal is clearer, the burrs are significantly reduced and the spectral amplitude of 4FSK signal is greatly improved. The success of this experiment suggests a strong promise of SR systems for digital signal detection in impact noise.

Stochastic resonance induced weak signal enhancement in a second-order tri-stable system with single-parameter adjusting

Weak signal detection methods based on stochastic resonance (SR) have been extensively studied due to their capability to utilize noise energy for enhancing weak signals. Among various SR models, the second-order tri-stable SR models have demonstrated their superiority in weak-signal detection with better output performance compared to other SR models. To optimize the output performance of the second-order tri-stable systems, a variety of parameter optimization methods have been proposed to optimize the parameters of the system. However, these optimization methods often require to optimize multiple parameters, which leads to an increase in computational costs and reduces the real-time processing efficiency of signal processing. Such multi-parameter optimization methods cannot meet the demands for timely signal processing in the context of big data. To address this challenge, this paper proposes two single-parameter-adjusting SR models. The proposed models can attain an ideal output performance by adjusting a single parameter in the SR system. The spectral amplification as an indicator is derived to quantitatively analyze the effects of the proposed models on SR output. On this basis, the influences of the proposed models on the SR output under different potential well parameters, noise intensities, signal frequency, and damping ratio are fully investigated through numerical simulations. At last, the proposed models are employed to process an experimental signal with a weak fault feature, and the experimental results verify the feasibility of the proposed models in optimizing the SR output. The research results can guide the design of tri-stable SR models and support the application of the SR-based signal processing model in the context of big data.

Magnetic Anomaly Detection Based on a Compound Tri-Stable Stochastic Resonance System.

In the case of strong background noise, a tri-stable stochastic resonance model has higher noise utilization than a bi-stable stochastic resonance (BSR) model for weak signal detection. However, the problem of severe system parameter coupling in a conventional tri-stable stochastic resonance model leads to difficulty in potential function regulation. In this paper, a new compound tri-stable stochastic resonance (CTSR) model is proposed to address this problem by combining a Gaussian Potential model and the mixed bi-stable model. The weak magnetic anomaly signal detection system consists of the CTSR system and judgment system based on statistical analysis. The system parameters are adjusted by using a quantum genetic algorithm (QGA) to optimize the output signal-to-noise ratio (SNR). The experimental results show that the CTSR system performs better than the traditional tri-stable stochastic resonance (TTSR) system and BSR system. When the input SNR is -8 dB, the detection probability of the CTSR system approaches 80%. Moreover, this detection system not only detects the magnetic anomaly signal but also retains information on the relative motion (heading) of the ferromagnetic target and the magnetic detection device.

An underdamped and delayed tri-stable model-based stochastic resonance

Stochastic resonance (SR) is investigated in an underdamped tri-stable potential system driven by Gaussian colored noise and a periodic excitation, where both displacement and velocity time-delayed states feedback are considered. It is challenging to study SR in a second-order delayed multi-stable system analytically. In this paper, the improved energy envelope stochastic average method is developed to derive the analytical expressions of stationary probability density (SPD) and spectral amplification. The effects of noise intensity, damping coefficient, and time delay on SR are analyzed. The results show that the shapes of joint SPD can be adjusted to the desired structure by choosing the time delay and feedback gains. For fixed time delay, the SR peak is increased for negative displacement or velocity feedback gain. Meanwhile, the SR peak is decreased while the optimal noise intensity increases with increasing correlation time of noise. The Monte Carlo simulations (MCS) confirm the effectiveness of the theoretical results.

On the shock performance of a hysteretic tri-stable vibration isolation system: Nonlinear phenomena and optimization

This work investigates the nonlinear response of a hysteretic multi-stable vibration isolation system under impulsive excitations. The multi-stable behavior is obtained by adding, in parallel with classical bearing devices, a mechanism with Negative Stiffness and superelastic hysteresis provided by Shape Memory Alloys (NS-SMA damping). By calibrating the design parameters, different types of stability can be achieved, such as bi-stability or tri-stability. The transition design parameter values between the different stability behaviors are identified analytically and define distinct space regions within the design parameters. The performances of the studied isolation system are numerically evaluated by computing the Shock Response Curves (SCR) and the performance maps are obtained by varying the design parameters and the amplitude of the motion. It is shown that the existence of two lateral wells in the potential profile can be exploited in order to trap the vibrating mass in them and effectively suppress accelerations and displacements. On the basis of this new observed phenomenon, renamed ‘well trapping’, the analytical equations for optimum parameter selection are obtained and furnished.

Piecewise time-delay tri-stable stochastic resonance system and its application in bearing fault diagnosis

Current studies of time-delay based systems are compared to classical tri-stable stochastic resonance (CTSR) system and do not reflect the effect of adding a delay term on unsaturated systems. In this paper, a piecewise delayed tri-stable stochastic resonance (PTTSR) system is proposed. Firstly, the equivalent Langevin for PTTSR system is derived and compared with the output of CTSR and piecewise tri-stable stochastic resonance (PTSR) systems, where the addition of the time-delay term can further improve the output amplitude. Secondly, mean first pass time (MFPT) and output signal-to-noise ratio (SNR) are derived, and the effects of different parameters on MFPT and SNR are investigated. increasing the feedback intensity and time-delay length improved the output SNR. Then, the input period signal is then numerically simulated using the signal-to-noise ratio gain (SNRG) as a measurement, which increases by 2.75 dB for PTTSR system compared to PTSR system. Finally, the three systems are used for bearing fault detection and the system parameters are optimized by genetic algorithm. The results showed that the output amplitude of PTTSR system is more than 12 times that of PTSR system and the SNRG increased by more than 2 dB. This demonstrates the superior performance of PTTSR system in detecting weak signals and provides good feasibility in engineering applications.