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Chaos analysis of sd oscillator with two-frequency excitation

NONLINEAR DYNAMICS 2024年第9期112卷 7649-7677页

作者： Peng, Ruyue Li, Qunhong Zhang, Wei Guangxi Univ Sch Phys Sci & Engn Technol Nanning 530004 Guangxi Peoples R China Guangxi Univ Coll Math & Informat Sci Nanning 530004 Guangxi Peoples R China Guangxi Univ Coll Civil Engn & Architecture Nanning 530004 Guangxi Peoples R China

Homoclinic bifurcation in a class of sd oscillators under external quasi-periodic excitation with two frequencies is investigated. By using the Melnikov method, the Melnikov function of the system under two-frequency excitation is derived, and the threshold condition of chaos in the system is obtained. Based on the threshold condition, then a complete description of the bifurcation sets and the chaotic regions in the parameter space are presented. The parameters in the chaotic regions are selected for numerical simulation, and the chaotic motion is verified by calculating the largest Lyapunov exponent of the system. In addition, due to the geometric strong nonlinearity of the sd oscillator, there are infinitely many extreme points in the frequency-dependent function of the Melnikov function. The previous research on the frequency-dependent function of the Melnikov function is mainly focused on one or two extremes. In this work, the dynamical system with infinitely many extreme points in the frequency-dependent function is considered, and a conjecture of chaotic region under this condition is given.

关键词： Quasiperiodic excitation Melnikov method Homoclinic bifurcation sd oscillator Chaos

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Research on dynamic characteristics of electromagnetic sd oscillator system under state feedback control

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INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS 2025年 176卷

作者： Wang, Meiqi Zu, Qian Wang, Ruichen Qin, Chengwei Liu, Pengfei Shijiazhuang Tiedao Univ Sch Mech Engn Shijiazhuang 050043 Peoples R China Shijiazhuang Tiedao Univ State Key Lab Mech Behav & Syst Safety Traff Engn Shijiazhuang 050043 Peoples R China Shijiazhuang Tiedao Univ Hebei Key Lab Mech Power & Transmiss Control Shijiazhuang 050043 Peoples R China CRRC Tangshan Co Ltd Tangshan 064000 Peoples R China

To enhance the robustness of the smooth and discontinuous (sd) oscillator system, an electromagnet incorporating displacement-velocity state feedback control is integrated above the oscillator. The electromagnetic interaction between the electromagnet and the oscillator facilitates accelerated system stabilization. This study investigates a novel electromagnetic sd oscillator. Initially, the steady-state amplitude-frequency and phasefrequency response characteristics are derived using the averaging method, with an analysis of how each state feedback parameter influences the amplitude-frequency curve. Utilizing Lyapunov stability theory and the Routh criterion, the stability conditions for the system's periodic solutions are established. Subsequently, the Melnikov theorem is employed to determine the necessary conditions for the onset of chaos within the system. Dynamic bifurcation analysis, maximum Lyapunov exponent (LLE) curves, and system behaviors under various parameters are examined to assess the impact of state feedback parameters on the chaotic boundary curve. On this basis, the global bifurcation characteristics of different parameters of the system are studied by using the cell mapping method, and the influence of each parameter on the number of attractors and the domain of attraction is analyzed. Finally, the displacement transmissibility is used to evaluate the vibration isolation effect of the system. The findings indicate that the electromagnetic sd oscillator exhibits superior vibration reduction capabilities, effectively mitigating chaos and bifurcation phenomena. Optimal selection of state feedback parameters enables rapid system stabilization under external excitation conditions.

关键词： sd oscillator Electromagnetic force Melnikov theory Dynamic bifurcation Lyapunov stability theory

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The effect of fractional damping and time-delayed feedback on the stochastic resonance of asymmetric sd oscillator

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NONLINEAR DYNAMICS 2022年第3期107卷 2099-2114页

作者： Wang, QiuBao Wu, Hao Yang, YueJuan Shijiazhuang Tiedao Univ Dept Math & Phys Shijiazhuang 050043 Hebei Peoples R China Tiangong Univ Sch Mech Engn Tianjin 300380 Peoples R China

This paper proposes the stiffness nonlinearities and asymmetric sd (smooth and discontinuous) oscillator under time-delayed feedback control with the fractional derivative damping. With the effect of displacement and velocity feedback, the oscillator can be adjusted to the desired vibration state, and then the stochastic resonance (SR) is achieved. This article discusses the contribution of various system parameters and time-delayed feedback to SR, especially which induced by fractional order. It should be noted that this paper provides effective guidance for fault diagnosis and weak signal detection, energy harvesting, vibration isolation and vibration reduction.

关键词： sd oscillator Fractional damping Time-delayed feedback Stochastic resonance

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Subharmonic resonance and fixed-range asymptotic stability of the fractional-order sd oscillator

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ADVANCES IN MECHANICAL ENGINEERING 2024年第1期16卷

作者： Wang, Cui-yan Wang, Ming-hao Chen, En-li Wang, Jun Shijiazhuang Tiedao Univ 17 Northeast Second Inner Ring Shijiazhuang 050043 Hebei Peoples R China Shijiazhuang Tiedao Univ State Key Lab Mech Behav & Syst Safety Traff Engn Shijiazhuang 050043 Peoples R China

The smooth and discontinuous (sd) oscillator is a typical system with strong nonlinear characteristics, and it is widely used in low-frequency vibration isolation and energy harvesting. A fractional damping model denoted by the Caputo model is introduced into the sd oscillator to adjust the property of the secondary resonance and evaluate the stability of the system. The influence of the fractional damping model on the one-third subharmonic resonance and the fixed-range asymptotic stability is studied. Residue theory and the Laplace transform are used to solve the fractional damping model. The amplitude-frequency response function and the existence conditions are derived by means of the averaging method. Lyapunov theory is used to determine the stable criteria of steady-state solutions. The cell-mapping method is ameliorated and used to calculate the fixed-range asymptotic stability of the one-third subharmonic resonance. The main results are as follows: a gap in the excitation amplitude occurs in the region of the existence condition of the one-third subharmonic resonance when the smooth parameter is smaller than 1. The generation of one-third subharmonic resonance is totally avoided for all frequencies when the excitation amplitudes are within the gap. The width of the gap, as well as the amplitude of the one-third subharmonic resonance, is affected by the parameters of the fractional damping term. The fixed-range asymptotic stability of the one-third subharmonic resonance is weak when the fractional damping parameters are large, which indicates a low resistance of the one-third subharmonic resonance to the external disturbance. The tuning effects of the fractional damping model on the one-third subharmonic resonance and fixed-range asymptotic stability are beneficial for the applications of sd oscillators.

关键词： sd oscillator nonlinear vibration subharmonic resonance nonlinear dynamics

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Bounded and Unbounded Solutions of the sd oscillator at Resonance

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数学理论与应用 2024年第4期44卷 45-69页

作者： Bian Jingke Liu Jie School of Mathematics and Information Science Henan Polytechnic UniversityJiaozuo 454003China

In this paper we study the boundedness and unboundedness of the solutions of the smooth and discontinuous(sd)oscillatorbegin{equation*}x''+f(x)x'+x-frac{x}{sqrt{x^{2}+alpha^{2}}}=p(t).end{equation*}Since f(x)≠0,the system is non-Hamiltonian,so we have to introduce some reversibility assumptions to apply a suitable twist theorem,for reversible maps with small ***,when the nonnegative parameterαdecreases to 0,the system becomes *** this case,we need to introduce some suitable transformations to overcome the lack of *** will prove that for any nonnegative parameterα,when p(t)is an odd periodic function satisfying∣∣∫2π0p(t)sintdift∣∣<4,all the solutions are bounded;when p(t)satisfies∣∣∫2π0p(t)sintdift∣∣>4,the sd oscillator has unbounded solutions,and when p(t)satisfies∣∣∫_(0)^(2π)p(t)sintdift∣∣≥4+|F|_(∞),all the solutions are unbounded.

关键词： sd oscillator Reversible system Bounded solution Unbounded solution

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Nonlinear dynamic response characteristics of sd oscillator with fractional damping

Nonlinear dynamic response characteristics of SD oscillator ...

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第八届国际振动工程会议

作者： Enli Chen Minghao Wang Meiqi Wang Yujian Chang State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures Shijiazhuang Tiedao University School of Mechanical Engineering Shijiazhuang Tiedao University School of Electrical and Electronic Engineering Shijiazhuang Tiedao University

The internal friction can be described more accurately by fractional damping than integral damping. The nonlinear dynamic response characteristics of sd oscillator with fractional damping under harmonic excitation are studied. The Fourier equivalent model is proposed to solve the nonlinear stiffness of the differential equation of system motion. The non-integrable nonlinear stiffness of the differential equation of the system is solved. The expression of amplitude-frequency response is obtained by solving the differential equation of system motion using the average method,and the stability of periodic solution is determined based on the Lyapunov stability theory and the Routh criterion. The results show the Fourier transform equivalent model of the nonlinear stiffness term of the sd oscillator can be applied to the motion characteristic of the system with large amplitude, which greatly improves the calculation accuracy. With the same damping coefficient, the amplitude-frequency response of the fractional damping system is different from that of the integral damping system, the resonance frequency and amplitude of the fractional damping system vary greatly. Changing the fractional coefficient will change the amplitude-frequency response backbone curve of the fractional damping system, but the integral damping system is not affected. When the fractional order is changed, the amplitude of the fractional damping system changes opposite on both sides of the cut-off point.

关键词： sd oscillator fractional damping backbone curve nonlinear vibration

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Global dynamics of a sd oscillator

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NONLINEAR DYNAMICS 2018年第3期91卷 1755-1777页

作者： Chen, Hebai Llibre, Jaume Tang, Yilei Fuzhou Univ Coll Math & Comp Sci Fuzhou 350002 Fujian Peoples R China Univ Autonoma Barcelona Dept Matemat E-08193 Barcelona Catalonia Spain Shanghai Jiao Tong Univ Sch Math Sci Shanghai 200240 Peoples R China

In this paper, we derive the global bifurcation diagrams of a sd oscillator which exhibits both smooth and discontinuous dynamics depending on the value of a parameter a. We research all possible bifurcations of this system, including Pitchfork bifurcation, degenerate Hopf bifurcation, homoclinic bifurcation, double limit cycle bifurcation, Bautin bifurcation and Bogdanov-Takens bifurcation. Besides, we show that the system has five limit cycles, including four small limit cycles and one large limit cycle. At last, we give all numerical phase portraits to illustrate our results.

关键词： sd oscillator Homoclinic loop Limit cycle Hopf bifurcation Bogdanov-Takens bifurcation Averaging method

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Delay-controlled primary and stochastic resonances of the sd oscillator with stiffness nonlinearities

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MECHANICAL SYSTEMS AND SIGNAL PROCESSING 2018年 103卷 216-235页

作者： Yang, Tao Cao, Qingjie Harbin Inst Technol Sch Astronaut Harbin 150001 Heilongjiang Peoples R China

This work presents analytical studies of the stiffness nonlinearities sd (smooth and discontinuous) oscillator under displacement and velocity feedback control with a time delay. The sd oscillator can capture the qualitative characteristics of quasi-zero-stiffness and negative-stiffness. We focus mainly on the primary resonance of the quasi-zero-stiffness sd oscillator and the stochastic resonance (SR) of the negative-stiffness sd oscillator. Using the averaging method, we have been analyzed the amplitude response of the quasi-zero-stiffness sd oscillator. In this regard, the optimum time delay for changing the control intensity according to the optimization standard proposed can be obtained. For the optimum time delay, increasing the displacement feedback intensity is advantageous to suppress the vibrations in resonant regime where vibration isolation is needed, however, increasing the velocity feedback intensity is advantageous to strengthen the vibrations. Moreover, the effects of time-delayed feedback on the SR of the negative stiffness sd oscillator are investigated under harmonic forcing and Gaussian white noise, based on the Langevin and Fokker-Planck approaches. The time-delayed feedback can enhance the SR phenomenon where vibrational energy harvesting is needed. This paper established the relationship between the parameters and vibration properties of a stiffness nonlinearities sd which provides the guidance for optimizing time-delayed control for vibration isolation and vibrational energy harvesting of the nonlinear systems. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： sd oscillator Stiffness nonlinearities Time-delayed feedback Primary resonance Stochastic resonance

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Nonlinear dynamics of the quasi-zero-stiffness sd oscillator based upon the local and global bifurcation analyses

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NONLINEAR DYNAMICS 2017年第2期87卷 987-1014页

作者： Hao, Zhifeng Cao, Qingjie Wiercigroch, Marian Harbin Inst Technol Sch Astronaut Ctr Nonlinear Dynam Res Harbin 150001 Peoples R China Univ Aberdeen Sch Engn Kings Coll Ctr Appl Dynam Res Aberdeen AB24 3UE Scotland

In this paper, the complicated nonlinear dynamics of the harmonically forced quasi-zero-stiffness sd (smooth and discontinuous) oscillator is investigated via direct numerical simulations. This oscillator considered that the gravity is composed of a lumped mass connected with a vertical spring of positive stiffness and a pair of horizontally compressed springs providing negative stiffness, which can achieve the quasi-zero stiffness widely used in vibration isolation. The local and global bifurcation analyses are implemented to reveal the complex dynamic phenomena of this system. The double-parameter bifurcation diagrams are constructed to demonstrate the overall topological structures for the distribution of various responses in parameter spaces. Using the Floquet theory and parameter continuation method, the local bifurcation patterns of periodic solutions are obtained. Moreover, the global bifurcation mechanisms for the crises of chaos and metamorphoses of basin boundaries are examined by analysing the attractors and attraction basins, exploring the evolutions of invariant manifolds and constructing the basin cells. Meanwhile, additional nonlinear dynamic phenomena and characteristics closely related to the bifurcations are discussed including the resonant tongues, jump phenomena, amplitude-frequency responses, chaotic seas, transient chaos, chaotic saddles, and also their generation mechanisms are presented.

关键词： sd oscillator QZS/HSLDS Local and global bifurcation analyses Floquet theory Numerical continuation Invariant manifold

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Harmonic and subharmonic solutions of the sd oscillator

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NONLINEAR DYNAMICS 2016年第4期84卷 2477-2486页

作者： Chen, Hebai Xie, Jianhua Southwest Jiaotong Univ Sch Mech & Engn Chengdu 610031 Sichuan Peoples R China

In this paper we study the existence and uniqueness of harmonic solutions and subharmonic ones of the smooth-and-discontinuous (sd, for short) differential equation , where and . However, for this oscillator it is eas... 详细信息

关键词： Harmonic solution Subharmonic solution sd oscillator

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