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Performance enhancement of a viscoelastic bistable energy harvester using time-delayed feedback control

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Chinese Physics B 2024年第6期33卷 142-154页

作者：黄美玲杨勇歌刘洋 School of Mathematics and Statistics Guangdong University of TechnologyGuangzhou 510520China State Key Laboratory for Strength and Vibration of Mechanical Structures Xi'an Jiaotong UniversityXi'an 710049China Exeter Small-Scale Robotics Laboratory Engineering DepartmentUniversity of ExeterNorth Park RoadExeter EX44QFUK

This paper focuses on the stochastic analysis of a viscoelastic bistable energy harvesting system under colored noise and harmonic excitation, and adopts the time-delayed feedback control to improve its harvesting efficiency. Firstly, to obtain the dimensionless governing equation of the system, the original bistable system is approximated as a system without viscoelastic term by using the stochastic averaging method of energy envelope, and then is further decoupled to derive an equivalent system. The credibility of the proposed method is validated by contrasting the consistency between the numerical and the analytical results of the equivalent system under different noise conditions. The influence of system parameters on average output power is analyzed, and the control effect of the time-delayed feedback control on system performance is compared. The output performance of the system is improved with the occurrence of stochastic resonance(SR). Therefore, the signal-to-noise ratio expression for measuring SR is derived, and the dependence of its SR behavior on different parameters is explored.

关键词： energy harvesting bistability stochastic averaging method stochastic resonance time-delayed feedback control

stochastic Bifurcation of an SIS Epidemic Model with Treatment and Immigration

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Journal of Applied Mathematics and Physics 2024年第6期12卷 2254-2280页

作者： Weipeng Zhang Dan Gu Department of Mathematics Jinan University Guangzhou China School of Mathematics and Statistics Northeast Normal University Changchun China

In this paper, we investigate an SIS model with treatment and immigration. Firstly, the two-dimensional model is simplified by using the stochastic averaging method. Then, we derive the local stability of the stochastic system by computing the Lyapunov exponent of the linearized system. Further, the global stability of the stochastic model is analyzed based on the singular boundary theory. Moreover, we prove that the model undergoes a Hopf bifurcation and a pitchfork bifurcation. Finally, several numerical examples are provided to illustrate the theoretical results. .

关键词： Epidemic Model stochastic averaging method Singular Boundary Theory stochastic Bifurcation

Response and Bifurcation of Fractional Duffing Oscillator under Combined Recycling Noise and Time-Delayed Feedback Control

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Wuhan University Journal of Natural Sciences 2023年第5期28卷 421-432页

作者： WANG Fang ZHANG Jiangang School of Mathematics and Physics Lanzhou Jiaotong UniversityLanzhou 730070GansuChina

Response and bifurcation of fractional Duffing oscillator under recycling noise and time-delayed feedback control are investigated. Firstly, based on the principle of the minimum mean square error and small time-delayed approximation and linearize the cubic stiffness term, the fractional derivative is equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Secondly, the Ito differential equations and one-dimensional Markov process are obtained according to stochastic averaging method, and the stochastic stability and stochastic bifurcation of the system are analyzed. Lastly, through joint probability density function diagram and the stationary probability density function diagram, the stochastic bifurcation behavior of system under the different time-delay, fractional order and noise intensity are discussed respectively, the validity of the theory and the occurrence of bifurcation phenomenon are verified.

关键词： recycling noise time-delay fractional derivative stochastic averaging method stochastic bifurcation

FIRST-PASSAGE TIME OF QUASI-NON-INTEGRABLE-HAMILTONIAN SYSTEM

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Acta Mechanica Sinica 2000年第2期16卷 183-192页

作者：甘春标徐博侯 Department of Mechanics. Zhejiang University Hangzhou China 310027

Studies on first-passage failure are extended to the multi-degree-of-freedom quasi-non-integrable-Hamiltonian systems under parametric excitations of Gaussian white noises in this paper. By the stochastic averaging method of energy envelope, the system's energy can be modeled as a one-dimensional approximate diffusion process by which the classical Pontryagin equation with suitable boundary conditions is applicable to analyzing the statistical moments of the first-passage time of an arbitrary order. An example is studied in detail and some numerical results are given to illustrate the above procedure.

关键词： Hamiltonian system non-integrable stochastic averaging method Pontryagin equation first-passage time

stochastic P-bifurcations of a noisy nonlinear system with fractional derivative element

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Acta Mechanica Sinica 2021年第3期37卷 507-515,I0004页

作者： Ya-Hui Sun Yong-Ge Yang Wei Xu School of Applied Mathematics Guangdong University of TechnologyGuangzhou 510520China Department of Applied Mathematics Northwestern Polytechnical UniversityXi'an 710072China

This paper investigates the stochastic P-bifurcation(SPB)of a fractionally damped oscillator subjected to additive and multiplicative Gaussian white *** transformation and the stochastic averaging technique are applied to derive the expression of probability density function(PDF)of the system *** conditions of the stochastic bifurcation induced by system parameters are presented based on the change in the number of extreme points of the probability density *** results are given to show the effectiveness of the proposed *** P-bifurcations for additive and multiplicative noise are studied in detail according to the critical conditions.

关键词： stochastic P-bifurcation Fractional derivative stochastic averaging method Gaussian white noise

Rate-dependent tipping-delay phenomenon in a thermoacoustic system with colored noise

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Science China(Technological Sciences) 2020年第11期63卷 2315-2327页

作者： ZHANG XiaoYu XU Yong LIU Qi KURTHS Jurgen Department of Applied Mathematics Northwestern Polytechnical UniversityXi’an 710072China MIIT Key Laboratory of Dynamics and Control of Complex Systems Northwestern Polytechnical UniversityXi’an 710072China Potsdam Institute for Climate Impact Research Potsdam 14412Germany

Tipping is a phenomenon in multistable systems where small changes in inputs cause huge changes in *** the parameter varies within a certain time scale,the rate will affect the tipping *** behaviors are undesirable in thermoacoustic systems,which are widely used in aviation,power generation and other ***,this paper aims at considering the tipping behaviors of the thermoacoustic system with the time-varying parameters and the combined excitations of additive and multiplicative colored *** dynamical behaviors for the proposed thermoacoustic model are implemented through the reduced Fokker-Planck-Kolmogorov equation derived by a standard stochastic averaging ***,the tipping problems of the rate-dependent thermoacoustic systems with random fluctuations are studied by virtue of the obtained probability density *** results show that the rate delays the value of the tipping parameter compared to the one with the quasi-steady assumption,which is called as a rate-dependent tipping-delay ***,the influences of the initial values,the rate,the changing time of the parameters,and the correlation time of the noises on the rate-dependent tipping-delay phenomenon are analyzed in *** results are of great significance for research in related fields such as aviation and land gas turbines.

关键词： thermoacoustic system colored noise rate-dependent tipping-delay phenomenon stochastic averaging method

stochastic averaging of quasi integrable and resonant Hamiltonian systems excited by fractional Gaussian noise with Hurst index 1/2

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Acta Mechanica Solida Sinica 2017年第1期30卷 11-19页

作者： Q.F.Lü M.L.Deng W.Q.Zhu Department of Mechanics State Key Laboratory of Fluid Power and Mechatronic SystemsKey Laboratory of Soft Machines and Smart Devices of Zhejiang ProvinceZhejiang University

A stochastic averaging method of quasi integrable and resonant Hamiltonian systems under excitation of fractional Gaussian noise （fGn） with the Hurst index 1/2 〈 H 〈 1 is proposed. First, the definition and the basic property of fGn and related fractional Brownian motion （iBm） are briefly introduced. Then, the averaged fractional stochastic differential equations （SDEs） for the first integrals and combinations of angle variables of the associated Hamiltonian systems are derived. The stationary probability density and statistics of the original systems are then obtained approximately by simulating the averaged SDEs numerically. An example is worked out to illustrate the proposed stochastic averaging method. It is shown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of original system agree well.

关键词： Quasi integrable and resonant Hamiltonian system Fractional Brownian motion Fractional Gaussian noise stochastic averaging method Internal resonant

Reliability and control of strongly nonlinear vibro-impact system under external and parametric Gaussian noises

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Science China(Technological Sciences) 2020年第9期63卷 1837-1845页

作者： LIU Li XU Wei YANG GuiDong HUANG DongMei Department of Applied Mathematics Northwestern Polytechnic UniversityXi’an710072China School of Mathematics and Statistics Ningxia UniversityYinchuan750021China School of Mathematics and Statistics Xidian UniversityXi’an710071China

This paper mainly focuses on reliability and the optimal bounded control for maximizing system reliability of a strongly nonlinear vibro-impact system. Firstly, the new stochastic averaging in which the impact condition is converted to the system energy is applied to obtain the averaged It? stochastic differential equation, by which the associated Backward Kolmogorov(BK)equation and Generalized Pontryagin(GP) equation are derived. Then, the dynamical programming equations are obtained based on the dynamical programming principle, by which the optimal bounded control for maximizing system reliability is ***, the effects of the bounded control and noise intensity on the reliability of the vibro-impact system are discussed in detail;meanwhile, the influence of impact conditions on the system's reliability is also studied. The feasibility and effectiveness of the proposed analytical method are substantiated by numerical results obtained from Monte-Carlo simulation.

关键词： vibro-impact system first-passage failure stochastic averaging method stochastic optimal control

stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation

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Chinese Physics B 2016年第2期25卷 13-21页

作者：杨勇歌徐伟孙亚辉谷旭东 Department of Applied Mathematics Northwestern Polytechnical University State Key Laboratory for Strength and Vibration of Mechanical Structures Xi'an Jiaotong University Department of Engineering Mechanics Northwestern Polytechnical University

This paper aims to investigate the stochastic response of the van der Pol （VDP） oscillator with two kinds of fractional derivatives under Gaussian white noise excitation. First, the fractional VDP oscillator is replaced by an equivalent VDP oscillator without fractional derivative terms by using the generalized harmonic balance technique. Then, the stochastic averaging method is applied to the equivalent VDP oscillator to obtain the analytical solution. Finally, the analytical solutions are validated by numerical results from the Monte Carlo simulation of the original fractional VDP oscillator. The numerical results not only demonstrate the accuracy of the proposed approach but also show that the fractional order, the fractional coefficient and the intensity of Gaussian white noise play important roles in the responses of the fractional VDP oscillator. An interesting phenomenon we found is that the effects of the fractional order of two kinds of fractional derivative items on the fractional stochastic systems are totally contrary.

关键词： stochastic averaging method fractional derivative van der Pol equivalent stochastic system