Bursting oscillation of a pendulum with irrational nonlinearity

Liu, C.; Jing, X.J.; Jiang, Wen-An; Ding, Hu; Chen, L.Q.; Bi, Qinsheng

doi:10.1016/j.ijnonlinmec.2022.104299

Cited by 5 publications

(3 citation statements)

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“…As a result, smooth or discontinuous dynamics, such as coexisting periodic motions and the transition of complex motion depending upon the value of a structural parameter, was demonstrated [12]. On this basis, Liu et al [13] explored a new bursting oscillation of this oscillator, employed three cases of excitation patterns for the investigation of the complex bursting behaviors and analyzed the principle behind the slow-bursting responses and the equilibrium stability via the method of multiple scales (MMS). Novel suspension vibration reduction systems with geometric nonlinear damping were put forward by the configuration of four springs with linear stiffness, and their performance for vibration isolation under impact and random excitations was then analyzed via numerical calculations [14].…”

Section: Introductionmentioning

confidence: 99%

Global Dynamics and Bifurcations of an Oscillator with Symmetric Irrational Nonlinearities

Liu,

Shang

2023

Fractal Fract

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This study’s objective is an irrationally nonlinear oscillating system, whose bifurcations and consequent multi-stability under the circumstances of single potential well and double potential wells are investigated in detail to further reveal the mechanism of the transition of resonance and its utilization. First, static bifurcations of its nondimensional system are discussed. It is found that variations of two structural parameters can induce different numbers and natures of potential wells. Next, the cases of mono-potential wells and double wells are explored. The forms and stabilities of the resonant responses within each potential well and the inter-well resonant responses are discussed via different theoretical methods. The results show that the natural frequencies and trends of frequency responses in the cases of mono- and double-potential wells are totally different; as a result of the saddle-node bifurcations of resonant solutions, raising the excitation level or frequency can lead to the coexistence of bistable responses within each well and cause an inter-well periodic response. Moreover, in addition to verifying the accuracy of the theoretical prediction, numerical results considering the disturbance of initial conditions are presented to detect complicated dynamical behaviors such as jump between coexisting resonant responses, intra-well period-two responses and chaos. The results herein provide a theoretical foundation for designing and utilizing the multi-stable behaviors of irrationally nonlinear oscillators.

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Section: Introductionmentioning

confidence: 99%

Global Dynamics and Bifurcations of an Oscillator with Symmetric Irrational Nonlinearities

Liu,

Shang

2023

Fractal Fract

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Section: Introductionmentioning

confidence: 99%

“…However, it should be pointed out that most studies have focused on nonlinear systems with rational polynomials, while very little research has been done on the fast-slow oscillations of systems exhibiting irrational nonlinearity [30]. In fact, the dynamical system with irrational nonlinearity is frequently encountered in pendulums [30,31], buckled beams [32] electro-mechanical system oscillators [33], energy harvesting devices [34], and other practical engineering. Recently, inspired by the elastic arch described by Thompson and Hunt [35], Cao et al [36] proposed an archetypal smooth and discontinuous (SD) oscillator with irrational nonlinearity, which is a simple mass-spring system constrained to a straight line by a geometrical parameter.…”

Section: Introductionmentioning

confidence: 99%

Fast-slow dynamics analysis in an externally excited smooth and discontinuous oscillator with a pair of irrational nonlinearities

Wei,

Han,

2023

Phys. Scr.

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The study of fast-slow oscillations in systems with irrational nonlinearity that may yield abundant dynamical mechanisms is not well developed. This paper aims to investigate the fast-slow dynamics in an excited mass-spring oscillator with a pair of irrational nonlinearities, which can undergo the dynamical transition from smooth to discontinuous characteristics depending on the values of a smoothness parameter. Three different types of fast-slow oscillations are reported in this interesting smooth and discontinuous (SD) oscillator with a pair of irrational nonlinearities. Due to the smooth and discontinuous characteristics of this SD oscillator, we consider its dynamical behaviors under the smooth and discontinuous cases, respectively. Based on the fast-slow analysis and the two-parameter bifurcation analysis, the smooth fast-slow dynamics associated with fold hysteresis and its turnover are revealed. In the discontinuous case, the system can be viewed as a piecewise-smooth dynamical system governed by three different subsystems in different regions divided by two nonsmooth boundaries. In particular, the nonsmooth boundaries can be divided into parts with different dynamical behaviors, including escaping and crossing lines. Unlike the smooth case, there is no change in the stability of the equilibrium in these three subsystems. However, transitions of system trajectory induced by crossing lines can account for the generation of fast-slow oscillations in the piecewise-smooth system. As a result, the smooth and piecewise-smooth fast-slow dynamics in the excited SD oscillator with a pair of irrational nonlinearities are revealed, which deepens the understanding of fast-slow dynamics of the dynamical systems with irrational nonlinearity.

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