Energy Flow Theory of Nonlinear Dynamical Systems with Applications

Xing, Jing Tang

doi:10.1007/978-3-319-17741-0

Cited by 13 publications

(17 citation statements)

References 129 publications

(170 reference statements)

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“…To investigate the behaviours of nonlinear dynamical systems, based on Eq. (1) in phase space, Xing [3] introduced three scalar energy flow variables:…”

Section: Energy Flow Variables and Equationsmentioning

confidence: 99%

“…Using Matlab software [3], the chaotic motion of nonlinear dynamical systems reported worldwide, such as the forced van der Pol system [7], Duffing system [8], Lorenz system [9, 10], Rössler system [11] and smooth and discontinuous (SD) oscillator [12][13][14] All the numerical results demonstrate that the energy characteristics of the chaotic motions of these nonlinear dynam-ical systems are correct. Here, as an example, a detailed discussion of the Lorenz system is presented, neglecting the related discussions for the other systems except listing the governing equations with the used parameters.…”

Section: Numerical Investigationmentioning

confidence: 99%

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Generalised energy conservation law for chaotic phenomena

Xing

2019

Acta Mech. Sin.

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Chaotic phenomena are increasingly being observed in all fields of nature, where investigations reveal that a natural phenomenon exhibits nonlinearities and attempts to reveal their deep underlying mechanisms. Chaos is normally understood as "a state of disorder", for which there is as yet no universally accepted mathematical definition. A commonly used concept states that, for a dynamical system to be classified as chaotic, it must have the following properties: be sensitive to initial conditions, show topological transitivity, have densely periodical orbits etc. Revealing the rules that govern chaotic motion is thus an important unsolved task for exploring nature. We present herein a generalised energy conservation law governing chaotic phenomena. Based on two scalar variables, viz. generalised potential and kinetic energies defined in the phase space describing nonlinear dynamical systems, we find that chaotic motion is periodic motion with infinite time period whose time-averaged generalised potential and kinetic energies are conserved over its time period. This implies that, as the averaging time is increased, the time-averaged generalised potential and kinetic energies tend to constants while the time-averaged energy flows, i.e., their rates of change with time, tend to zero. Numerical simulations on reported chaotic motions, such as the forced van der Pol system, forced Duffing system, forced smooth and discontinuous oscillator, Lorenz's system, and Rössler's system, show the above conclusions to be correct according to the results presented herein. This discovery may indicate that chaotic phenomena in nature could be controlled because, even though their instantaneous states are disordered, their long-time averages can be predicted.

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“…To investigate the behaviours of nonlinear dynamical systems, based on Eq. (1) in phase space, Xing [3] introduced three scalar energy flow variables:…”

Section: Energy Flow Variables and Equationsmentioning

confidence: 99%

Section: Numerical Investigationmentioning

confidence: 99%

Generalised energy conservation law for chaotic phenomena

Xing

2019

Acta Mech. Sin.

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“…Following the energy flow theory [38] and pre-multiplying Eq. (9) by the velocity vectorŻ T , we can derive the energy flow equation of the system in the forṁ…”

Section: Energy Flow Equationmentioning

confidence: 99%

“…(9), of which the solutions are complex and their corresponding imaginary parts are the physical variables. As indicated in the book [38], when calculating powers by Eqs. (10) and (18), you have to use the real physical variables to get their correct results.…”

Section: Mode Summationmentioning

confidence: 99%

Dynamic interactions of an integrated vehicle–electromagnetic energy harvester–tire system subject to uneven road excitations

et al. 2017

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An investigation is undertaken of an integrated mechanical-electromagnetic coupling system consisting of a rigid vehicle with heave, roll, and pitch motions, four electromagnetic energy harvesters and four tires subject to uneven road excitations in order to improve the passengers' riding comfort and harvest the lost engine energy due to uneven roads. Following the derived mathematical formulations and the proposed solution approaches, the numerical simulations of this interaction system subject to a continuous sinusoidal road excitation and a single ramp impact are completed. The simulation results are presented as the dynamic response curves in the forms of the frequency spectrum and the time history, which reveals the complex interaction characteristics of the system for vibration reductions and energy harvesting performance. It has addressed the coupling effects on the dynamic characteristics of the integrated system caused by: (1) the natural modes and frequencies of the vehicle; (2) the vehicle rolling and pitching motions; (3) different road excitations on four wheels; (4) the time delay of a road ramp to impact both the front and rear wheels, etc., which cannot be tackled by an often used quarter vehicle model. The guidelines for engineering applications are given. The developed coupling model and the revealed concept provide a means with analysis idea to investigate the details of four energy harvester motions for electromagnetic suspension designs in order to replace the current passive vehicle isolators and to harvest the lost engine energy. Potential further research directions are suggested for readers to consider in the future. B Jing Tang Xing

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“…Power flow could be used to quantify the isolation performance and reflect more information of the dynamical system [1][2][3][4][5][6]. In many cases, the mounting base was flexible.…”

Section: Introductionmentioning

confidence: 99%

Power Flow in a Two-Stage Nonlinear Vibration Isolation System with High-Static-Low-Dynamic Stiffness

Shao

Ding

2018

Shock and Vibration

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The manuscript concerns the power flow characterization in a two-stage nonlinear vibration isolator comprising three springs, which are configured so that each stage of the system has a high-static-low-dynamic stiffness. To demonstrate the distinction of evaluation for vibration isolation using power flow, force transmissibility is used for comparison. The dynamic behavior of the isolator subject to harmonic excitation, however, is of interest here. The harmonic balance method (HBM) could be used to analyze the frequency response curve (FRC) of the strong nonlinear vibration system. A suggested stability analysis to distinguish the stable and the unstable HBM solutions is described. Increasing both upper and lower nonlinear stiffness could bend the first resonant peak to the left. The isolation range in the power and the force transmissibility plot could be extended to the lower frequencies when the nonlinear stiffness is increased, but the rate of roll-off for the power transmissibility is twice the rate for the force transmissibility at each horizontal stiffness setting. An explanation for this phenomenon is given in the paper.

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Energy Flow Theory of Nonlinear Dynamical Systems with Applications

Cited by 13 publications

References 129 publications

Generalised energy conservation law for chaotic phenomena

Generalised energy conservation law for chaotic phenomena

Dynamic interactions of an integrated vehicle–electromagnetic energy harvester–tire system subject to uneven road excitations

Power Flow in a Two-Stage Nonlinear Vibration Isolation System with High-Static-Low-Dynamic Stiffness

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