Energy transfer in double plate system dynamics

The study of energy transfer between coupled subsystems in a hybrid system is very important for applications. This paper presents an analytical analysis of energy transfer between plates of a visco-elastically connected double-plate system in free transversal vibrations. The analytical analysis shows that the visco-elastic connection between plates is responsible for the appearance of two-frequency regime in the time function, which corresponds to one eigen amplitude function of one mode, and also that time functions of different vibration modes are uncoupled, but energy transfer between plates in one eigen mode appears. It was shown for each shape of vibrations. Series of the two Lyapunov exponents corresponding to the one eigen amplitude mode are expressed by using the energy of the corresponding eigen amplitude time component.     

Energy transfer in the hybrid system dynamics (energy transfer in the axially moving double belt system)

First, as an introduction, using the author’s published references, a short survey of an analytical study of the energy transfer between two coupled subsystems, as well as between a linear and nonlinear oscillators of a hybrid system, in the free and forced vibrations of a different type of inter connections between subsystems is presented. Second, as author’s new research result, an analytical study of the energy transfer between two coupled like-string belts interconnected by light pure elastic layer in the axially moving sandwich double belt system, in the free vibrations is presented. On the basis of the obtained analytical expressions for the kinetic and potential energy of the belts and potential energy of the of light pure elastic distributed layer numerous conclusions are derived. In the pure linear elastic double belt system no transfer energy between different eigen modes of transversal vibrations of the axially moving double belt system, but in every from of the set of the infinite numbers eigen modes, there are transfer energy between belts. Each of the eigen modes of the free transversal vibrations are like two-frequency. The change of the potential energy of the booth belts is four frequency, and interaction part of the potential energy is one frequency in the each eigen mode. Changes of the kinetic energy of the both belts of the sandwich double axially moving bet system is two frequency like oscillatory regimes with two time multiplicities of the eineg frequencies of the corresponding eigen amplitude mode.     

Complex dynamics and targeted energy transfer in linear oscillators coupled to multi-degree-of-freedom essentially nonlinear attachments

We study the dynamics of a system of coupled linear oscillators with a multi-DOF end attachment with essential (nonlinearizable) stiffness nonlinearities. We show numerically that the multi-DOF attachment can passively absorb broadband energy from the linear system in a one-way, irreversible fashion, acting in essence as nonlinear energy sink (NES). Strong passive targeted energy transfer from the linear to the nonlinear subsystem is possible over wide frequency and energy ranges. In an effort to study the dynamics of the coupled system of oscillators, we study numerically and analytically the periodic orbits of the corresponding undamped and unforced hamiltonian system with asymptotics and reduction. We prove the existence of a family of countable infinity of periodic orbits that result from combined parametric and external resonance interactions of the masses of the NES. We numerically demonstrate that the topological structure of the periodic orbits in the frequency–energy plane of the hamiltonian system greatly influences the strength of targeted energy transfer in the damped system and, to a great extent, governs the overall transient damped dynamics. This work may be regarded as a contribution towards proving the efficacy the utilizing essentially nonlinear attachments as passive broadband boundary controllers. PACS numbers: 05.45.Xt, 02.30.Hq     

Heteroclinic motion and energy transfer in coupled oscillator with nonlinear magnetic coupling

In this paper, we consider two coupled oscillators exhibiting both transient chaos and energy transfer from mechanical to electrical oscillators. Melnikov method is applied to these oscillators with linear damping and strongly nonlinear coupling terms in order to study the possibility of existence of chaos and transversal heteroclinic orbits and their control in a dynamical system. The energy transfer is studied using a qualitative measure of the system which can be obtained by computing the energy dissipated in it. At last, the numerical simulation is carried out for this system.     

Transitions from strongly to weakly-nonlinear dynamics in a class of exactly solvable oscillators and nonlinear beat phenomena

In this paper, a regular perturbation tool is suggested to bridge the gap between weakly and strongly nonlinear dynamics based on exactly solvable oscillators with trigonometric characteristics considered by Nesterov (Proc. Mosc. Inst. Power Eng. 357:68–70, 1978). It is shown that the corresponding action-angle variables linearize the original oscillators with no special functions involved. As a result, linear and strongly nonlinear areas of the dynamics are described within the same perturbation procedure. The developed tool is applied then to analyzing the nonlinear beat and energy localization phenomena in two linearly coupled Duffing oscillators. It is shown that the principal phase variable describing the beat phenomena is governed by the hardening Nesterov oscillator with some perturbation due to qubic nonlinearity and coupling between the oscillators. As a result, the above class of strongly nonlinear oscillators is given clear physical meaning, whereas a closed form analytical solution is obtained for nonlinear beat and localization dynamics. Based on this solution, necessary and sufficient conditions for onset of energy localization are obtained.     

Dynamic instabilities in coupled oscillators induced by geometrically nonlinear damping

The dynamics of a system of coupled oscillators possessing strongly nonlinear stiffness and damping is examined. The system consists of a linear oscillator coupled to a strongly nonlinear, light attachment, where the nonlinear terms of the system are realized due to geometric effects. We show that the effects of nonlinear damping are far from being purely parasitic and introduce new dynamics when compared to the corresponding systems with linear damping. The dynamics is analyzed by performing a slow/fast decomposition leading to slow flows, which in turn are used to study transient instability caused by a bifurcation to 1:3 resonance capture. In addition, a new dynamical phenomenon of continuous resonance scattering is observed that is both persistent and prevalent for the case of the nonlinearly damped system: For certain moderate excitations, the transient dynamics “tracks” a manifold of impulsive orbits, in effect transitioning between multiple resonance captures over definitive frequency and energy ranges. Eventual bifurcation to 1:3 resonance capture generates the dynamic instability, which is manifested as a sudden burst of the response of the light attachment. Such instabilities that result in strong energy transfer indicate potential for various applications of nonlinear damping such as in vibration suppression and energy harvesting.     

Nonlinear normal modes of a self-excited system driven by parametric and external excitations

Vibrations of nonlinear coupled parametrically and self-excited oscillators driven by an external harmonic force are presented in the paper. It is shown that if the force excites the system inside the principal parametric resonance then for a small excitation amplitude a resonance curve includes an internal loop. To find the analytical solutions, the problem is reduced to one degree of freedom oscillators by applications of Nonlinear Normal Modes (NNMs). The NNMs are formulated on the basis of free vibrations of a nonlinear conservative system as functions of amplitude. The analytical results are validated by numerical simulations and an essential difference between linear and nonlinear modes is pointed out.     

Lyapunov function construction for nonlinear stochastic dynamical systems

Though the Lyapunov function method is more efficient than the largest Lyapunov exponent method in evaluating the stochastic stability of multi-degree-of-freedom (MDOF) systems, the construction of Lyapunov function is a challenging task. In this paper, a specific linear combination of subsystems’ energies is proposed as Lyapunov function for MDOF nonlinear stochastic dynamical systems, and the corresponding sufficient condition for the asymptotic Lyapunov stability with probability one is then determined. The proposed procedure to construct Lyapunov function is illustrated and validated with several representative examples, where the influence of coupled/uncoupled dampings and excitation intensities on stochastic stability is also investigated.     

Using Fourier differential quadrature method to analyze transverse nonlinear vibrations of an axially accelerating viscoelastic beam

In this paper, a Fourier expansion-based differential quadrature (FDQ) method is developed to analyze numerically the transverse nonlinear vibrations of an axially accelerating viscoelastic beam. The partial differential nonlinear governing equation is discretized in space region and in time domain using FDQ and Runge–Kutta–Fehlberg methods, respectively. The accuracy of the proposed method is represented by two numerical examples. The nonlinear dynamical behaviors, such as the bifurcations and chaotic motions of the axially accelerating viscoelastic beam, are investigated using the bifurcation diagrams, Lyapunov exponents, Poincare maps, and three-dimensional phase portraits. The bifurcation diagrams for the in-plane responses to the mean axial velocity, the amplitude of velocity fluctuation, and the frequency of velocity fluctuation are, respectively, presented when other parameters are fixed. The Lyapunov exponents are calculated to further identify the existence of the periodic and chaotic motions in the transverse nonlinear vibrations of the axially accelerating viscoelastic beam. The conclusion is drawn from numerical simulation results that the FDQ method is a simple and efficient method for the analysis of the nonlinear dynamics of the axially accelerating viscoelastic beam.     

The tuned bistable nonlinear energy sink

A bistable nonlinear energy sink conceived to mitigate the vibrations of host structural systems is considered in this paper. The hosting structure consists of two coupled symmetric linear oscillators (LOs), and the nonlinear energy sink (NES) is connected to one of them. The peculiar nonlinear dynamics of the resulting three-degree-of-freedom system is analytically described by means of its slow invariant manifold derived from a suitable rescaling, coupled with a harmonic balance procedure, applied to the governing equations transformed in modal coordinates. On the basis of the first-order reduced model, the absorber is tuned and optimized to mitigate both modes for a broad range of impulsive load magnitudes applied to the LOs. On the one hand, for low-amplitude, in-well, oscillations, the parameters governing the bistable NES are tuned in order to make it functioning as a linear tuned mass damper (TMD); on the other, for high-amplitude, cross-well, oscillations, the absorber is optimized on the basis of the invariant manifolds features. The analytically predicted performance of the resulting tuned bistable nonlinear energy sink (TBNES) is numerically validated in terms of dissipation time; the absorption capabilities are eventually compared with either a TMD and a purely cubic NES. It is shown that, for a wide range of impulse amplitudes, the TBNES allows the most efficient absorption even for the detuned mode, where a single TMD cannot be effective.     

Transient in a two-DOF nonlinear system

Dynamics of a system containing a linear oscillator, linearly coupled to an essentially nonlinear attachment, is considered. A damping is taken into account. It is assumed that some initial excitation implies vibrations of the linear oscillator. Envelops of the subsystem’s kinetic energies are selected to use the numerical investigation of transient in the system. The parametrical optimization approach is used to obtain regions of effective energy transfer in the system parameter space. It is demonstrated that this efficient energy transfer may be obtained for a rather small value of the attachment mass.     

Periodic orbits,basins of attraction and chaotic beats in two coupled Kerr oscillators

Kerr oscillators are model systems which have practical applications in nonlinear optics. Optical Kerr effect, i.e., interaction of optical waves with nonlinear medium with polarizability χ ⁽³⁾ is the basic phenomenon needed to explain, for example, the process of light transmission in fibers and optical couplers. In this paper, we analyze the two Kerr oscillators coupler and we show that there is a possibility to control the dynamics of this system, especially by switching its dynamics from periodic to chaotic motion and vice versa. Moreover, the switching between two different stable periodic states is investigated. The stability of the system is described by the so-called maps of Lyapunov exponents in parametric spaces. Comparison of basins of attractions between two Kerr couplers and a single Kerr system is also presented.     

Dynamics of coupled nonlinear oscillators: The formation of long-lived vibrational states in the case of molecular systems

Nonlinear vibrations in a closed system of coupled nonlinear oscillators are studied using acetylene type molecules as an example. A criterion for the stable existence of long-lived vibrational states—local modes—in one of the oscillators is obtained. It is shown that the disappearance of a local mode, as well as its appearance, proceeds abruptly, and the mechanism of stabilization of these excitations is due to the presence or absence of internal resonances of an oscillatory system such as any polyatomic molecule. Energy values needed to excite vibrations in which local modes can appear are determined. It is shown that calculation results agree with experimental data.     

Local sensitivity of spatiotemporal structures

We present an index for the local sensitivity of spatiotemporal structures in coupled oscillatory systems based on the properties of local-in-space, finite-time Lyapunov exponents. For a system of nonlocally coupled Rössler oscillators, we show that variations of this index for different oscillators reflect the sensitivity to noise and the onset of spatial chaos for the patterns where coherence and incoherence regions coexist.     

Modeling a synthetic biological chaotic system: relaxation oscillators coupled by quorum sensing

Chaos exists in biological systems. Through investigating synthetic genetic relaxation oscillators coupled by quorum sensing, this paper reports a chaotic system. The detailed dynamical behaviors of this chaotic biological system are investigated, including Lyapunov exponents spectrum, bifurcation, and Poincaré mapping.     

Estimation of Lyapunov exponents from a time series for n-dimensional state space using nonlinear mapping

It has been demonstrated that when estimating Lyapunov exponents using a time series, nonlinear mapping used for characterizing the evolution of the neighbors leads to more accurate negative exponents and is more robust to noise in the times series. However, the number of unknown elements of the matrices associated with nonlinear mapping increases significantly with the embedding dimensions of the state space where the dynamics is reconstructed. Such unknown coefficients are solved from a set of linear algebraic equations based on the least square-root fit method. Derivation of such linear equations and computer programming are tedious and error prone especially for the systems with high embedding dimensions. In this work, we develop a general form of the linear algebraic equations and the corresponding computer program in terms of arbitrary embedding dimensions. A stable robotic system with all negative Lyapunov exponents and the Lorenz system with positive, zero, and negative exponents are used to demonstrate the efficacy of the proposed method. The work can contribute significantly to estimating Lyapunov exponents for systems with large embedding dimensions.     

Localization of energy in nonlinear systems with two degrees of freedom

Energy pumping in a two-degrees-of-freedom system with linear and essentially nonlinear oscillators is studied. The kinetic energy envelopes of the linear and nonlinear subsystems are chosen to be the main characteristics of the process under consideration. A criterion that the energy of the linear oscillator excited at time zero is completely pumped over into the nonlinear oscillator is established together with an additional condition whereby the energy does not return to the linear subsystem. Optimal energy pumping mode is established using global optimization. The effect of the parameters of the system on the main characteristics is assessed __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 115–125, May 2007.     

Generating tri-chaos attractors with three positive Lyapunov exponents in new four order system via linear coupling

This paper presents a new hyperchaotic system with three positive Lyapunov exponents (called Tri-Chaos). Via linear coupling, Mathieu, and van der Pol systems are coupled with each other and then become a new four order system??Mathieu?Cvan der Pol autonomous system. As we know, two positive Lyapunov exponents confirm hyperchaotic nature of its dynamics and it means that the system can present more complicated behavior than ordinary chaos. We further generate three positive Lyapunov exponents in a new coupled nonlinear system and anticipate the advanced application in secure communication. Not only a new chaotic system with three Lyapunov exponents is proposed, but also its implementation of an electronic circuit is put into practice in this article. The phase portrait, electronic circuit, power spectrum, Lyapunov exponents, and 2-D and 3-D parameter diagram of tri-chaos with three positive Lyapunov exponents of the new system will be shown in this paper.     

双状态切换下BVP振子的复杂行为分析

非线性切换系统具有广泛的工程背景,而传统的非线性理论不能直接用来解决其中的问题,因而成为当前国内外热点和前沿课题之一. 目前相关工作大都是围绕固定时间或单状态切换开展的,而实际工程系统大都属于多状态切换问题,同时多状态切换涉及到更为丰富的动力学行为. 本文基于两广义BVP 振子,通过引入双向切换开关,构建了双状态切换下的非线性动力学模型,进而研究状态切换导致的各种运动模式及其相应的产生机制. 应用非光滑系统的Poincaré映射理论,推导了双状态切换下的Lyapunov 指数的计算公式,结合子系统的分岔分析,得到了切换系统随分岔参数变化的动力学演化过程及其相应的最大Lyapunov 指数的变化情况. 得到了双状态切换条件下系统存在着各种形式的振荡行为,分析了诸如周期突变等现象及通往混沌的倍周期分岔道路,揭示了不同运动模式的产生机制及倍周期序列的本质. 与固定时间切换和单状态切换系统不同,双临界状态切换系统存在着更为丰富的非线性现象,其主要原因在于双状态切换会产生更多的切换点,且切换点的位置更加多变. 同时切换系统的倍周期分岔序列与光滑系统中的倍周期分岔序列不同,切换系统的倍周期分岔序列只对应于切换点数目的成倍增加,而其相应的周期一般不对应于严格的周期倍化过程.