Dynamic Analysis of the Smooth-and-Discontinuous Oscillator under Constant Excitation

Tian, Ruilan; Wu, Qiliang; Liu, Zhong-Jia; Yang, Xinwei

doi:10.1088/0256-307x/29/8/084706

Cited by 14 publications

(2 citation statements)

References 14 publications

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“…Many complex nonlinear dynamic behaviors of the SD oscillator have been proven by Tian et al, such as Hopf bifurcation, 22 codimension-2 bifurcation, 24 complex KAM structures, and chaotic behaviors. 25 Santhosh et al 26 analyzed the numerical solutions of an SD oscillator from the perspective of the frequency domain. Chen et al 27 found the sufficient and necessary existence conditions and the number of limit circles of grazing bifurcations, as well as the subharmonic solutions and harmonic solutions.…”

Section: Introductionmentioning

confidence: 99%

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

Cui-yan,

Ming-hao,

En-li

et al. 2024

Advances in Mechanical Engineering

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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.

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Section: Introductionmentioning

confidence: 99%

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

Cui-yan,

Ming-hao,

En-li

et al. 2024

Advances in Mechanical Engineering

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“…Fork-type bifurcation, single Hopf bifurcation and double Hopf bifurcation, homoclinic bifurcation, and closed-orbit bifurcation around the equilibrium points were discussed in (Cao et al, 2008a,b,c;Tian et al, 2010a,b). The response of the SD oscillator was derived under constant excitation (Tian et al, 2012). Han et al (2012) and Cao et al (2012) studied the strong irrational nonlinear oscillator stability and bifurcation.…”

Section: Introductionmentioning

confidence: 99%

Experimental study of the nonlinear dynamics of a smooth and discontinuous oscillator with different smoothness parameters and initial values

Chang

Chen

Feng³

2019

Journal of Theoretical and Applied Mechanics

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The dynamic response of a nonlinear system is very sensitive to initial conditions. Both the irrational nonlinearity and the large displacement of a smooth and discontinuous (SD) oscillator have been studied in this paper. An experimental study has been conducted on a model of the SD oscillator with different initial conditions and smoothness parameters. Experimental results indicate that tiny variation in the initial displacement will lead to different kinds of vibrations, and the system exhibits a wide range of nonlinear dynamical phenomena with the change of smoothness parameters. All experimental results are in good conformity with numerical simulation results.

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

Wang

Yang

2022

Nonlinear Dyn

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This paper proposes the stiffness nonlinearities and asymmetric SD (smooth and discontinuous) oscillators under time-delay feedback control with a fractional 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-delay feedback to SR, especially which induced by fractional damping. It should be noted that this paper provides effective guidance for fault diagnosis and weak signal detection, energy harvesting, vibration isolation and vibration reduction.

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Dynamic Analysis of the Smooth-and-Discontinuous Oscillator under Constant Excitation

Cited by 14 publications

References 14 publications

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

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

Experimental study of the nonlinear dynamics of a smooth and discontinuous oscillator with different smoothness parameters and initial values

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

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