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Sliding bifurcations of a self-excited sd oscillator with nonlinear damping

Chaos, Solitons and Fractals 2025年 198卷

作者： Fu, Shihui Páez Chávez, Joseph Liu, Yang School of Mathematics and Statistics Zhengzhou University Zhengzhou Henan 450001 China Center for Applied Dynamical Systems and Computational Methods (CADSCOM) Faculty of Natural Sciences and Mathematics Escuela Superior Politécnica del Litoral P.O. Box 09-01-5863 Guayaquil Ecuador Center for Dynamics Department of Mathematics TU Dresden Dresden D-01062 Germany Exeter Small-Scale Robotics Laboratory Engineering Department University of Exeter Exeter EX4 4QF United Kingdom

This paper investigates the sliding bifurcations of a self-excited smooth and discontinuous (sd) oscillator with a nonlinear damper driven by moving belt friction. A generalized Filippov framework is applied to derive an explicit relationship between the coefficient of the nonlinear term and the maximum static friction coefficient. The conditions for the existence of various sliding bifurcations, including crossing-sliding, grazing-sliding, switching-sliding, and adding-sliding bifurcations, are established. The theoretical analysis is validated through numerical simulations, which confirm the presence of sliding bifurcations and highlight their impact on the system dynamics. To further examine the nonlinear system response, a numerical study via continuation methods is conducted. Bifurcation diagrams are generated to track the evolution of system behavior as key parameters vary. The results reveal complex dynamical transitions, including the onset of sliding bifurcations and their influence on the stability of periodic orbits. The findings demonstrate that nonlinear damping plays a crucial role in shaping the bifurcation structure and sliding behavior of the oscillator. This study provides a refined theoretical framework for analyzing sliding bifurcations in non-smooth dynamical systems and offers insights into the control and design of friction-driven oscillators with nonlinear damping. © 2025 The Authors

关键词： Dry friction General Filippov theory sd oscillator Sliding bifurcation Sliding region

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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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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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Chaos analysis of sd oscillator with two-frequency excitation

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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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The codimension-two bifurcation for the recent proposed sd oscillator

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NONLINEAR DYNAMICS 2010年第1-2期59卷 19-27页

作者： Tian, Ruilan Cao, Qingjie Yang, Shaopu Ctr Nonlinear Dynam Res Shijiazhang Railway Inst Shijiazhuang 050043 Peoples R China

In this paper, the complicated nonlinear dynamics at the equilibria of sd oscillator, which exhibits both smooth and discontinuous dynamics depending on the value of a parameter alpha, are investigated. It is found that sd oscillator admits codimension-two bifurcation at the trivial equilibrium when alpha=1. The universal unfolding for the codimension-two bifurcation is also found to be equivalent to the damped sd oscillator with nonlinear viscous damping. Based on this equivalence between the universal unfolding and the damped system, the bifurcation diagram and the corresponding codimension-two bifurcation structures near the trivial equilibrium are obtained and presented for the damped sd oscillator as the perturbation parameters vary.

关键词： sd oscillator Codimension-two bifurcation Universal unfolding Melnikov method Homoclinic orbit

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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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IRRATIONAL ELLIPTIC FUNCTIONS AND THE ANALYTICAL SOLUTIONS OF sd oscillator

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JOURNAL OF THEORETICAL AND APPLIED MECHANICS 2012年第3期50卷 701-715页

作者： Cao, Qingjie Wang, Dan Chen, Yushu Wiercigroch, Marian Harbin Inst Technol Sch Astronaut Ctr Nonlinear Dynam Res Harbin 150006 Peoples R China Shijiazhuang Tiedao Univ Ctr Nonlinear Dynam Res Shijiazhuang Peoples R China Univ Aberdeen Kings Coll Dept Engn Ctr Appl Dynam Res Aberdeen Scotland

The smooth and discontinuous (sd) oscillator is a strongly nonlinear system with an irrational restoring force proposed in P.R.E (2006), which leads to barriers for the conventional methods to investigate the dynamical behaviour directly. In this paper, two kinds of irrational elliptic functions and a kind of hyperbolic functions are defined in the real domain to formulate the analytical solutions of the system. The properties of the functions are obtained including differentiability, periodicity and parity. As the application of the defined irrational functions, the chaotic thresholds of the oscillator are also depicted by using the Melnikov method. Numerical analysis shows the efficiency of the proposed procedure.

关键词： sd oscillator irrational nonlinearity irrational elliptic functions threshold of chaos

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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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Modelling and experimental investigations on the geometrical nonlinear dynamics of the sd oscillator

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INTERNATIONAL JOURNAL OF MODELLING IDENTIFICATION AND CONTROL 2016年第3期25卷 190-198页

作者： Chen, Enli Chu, Bogang Wang, Yang Shijiazhuang Tiedao Univ Sch Mech Engn Shijiazhuang 050043 Hebei Peoples R China Shijiazhuang Tiedao Univ Sch Civil Engn Shijiazhuang 050043 Hebei Peoples R China

In this paper, an experimental rig and a dynamic model for the smooth and discontinuous (sd) oscillator is set up to describe the geometrical nonlinearity caused by large deformation, which is typical of irrational nonlinearity with double stabilities. The system parameters of the proposed device are adjustable and measurable during the experiment for different working conditions. Various experimental investigations under the harmonic excitation are carried out to explore the complex nonlinear behaviour of periodic solution, bifurcation phenomenon, chaotic phenomena and the transitions among the multiple motion patterns. Numerical simulations are carried out to demonstrate the reliability of the proposed experimental rig, showing a good agreement with presented experimental results. The results presented via experimental investigations reflect the natural behaviour of the geometrical nonlinear dynamics.

关键词： sd oscillator nonlinear behaviour experimental parameter recognition nonlinear experiment rig geometrical nonlinearity irrational nonlinearity bifurcation chaos spectrum Poincare section

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