On the response of a harmonically excited two degree-of-freedom system consisting of a linear and a nonlinear quasi-zero stiffness oscillator

There are many systems which consist of a nonlinear oscillator attached to a linear system, examples of which are nonlinear vibration absorbers, or nonlinear systems under test using shakers excited harmonically with a constant force. This paper presents a study of the dynamic behaviour of a specific two degree-of-freedom system representing such a system, in which the nonlinear system does not affect the vibration of the forced linear system. The nonlinearity of the attachment is derived from a geometric configuration consisting of a mass suspended on two springs which are adjusted to achieve a quasi-zero stiffness characteristic with pure cubic nonlinearity. The response of the system at the frequency of excitation is found analytically by applying the method of averaging. The effects of the system parameters on the frequency-amplitude response of the relative motion are examined. It is found that closed detached resonance curves lying outside or inside the continuous path of the main resonance curve can appear as a part of the overall amplitude-frequency response. Two typical situations for the creation of the detached resonance curve inside the main resonance curve, which are dependent on the damping in the nonlinear oscillator, are discussed.     

Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators

We show that a hyperbolic chaos can be observed in resonantly coupled oscillators near a Hopf bifurcation, described by normal-form-type equations for complex amplitudes. The simplest example consists of four oscillators, comprising two alternatively activated, due to an external periodic modulation, pairs. In terms of the stroboscopic Poincaré map, the phase differences change according to an expanding Bernoulli map that depends on the coupling type. Several examples of hyperbolic chaos for different types of coupling are illustrated numerically.     

The vibro-impact response of a harmonically excited and preloaded one-dimensional linear oscillator

The non-linear response of a one-dimensional oscillator preloaded against a stop and subjected to harmonic excitation is investigated. The analysis includes a discussion of the existence and stability of subharmonic single impact responses and a comparison between the preloaded case and impacting with a clearance.     

Forced nonlinear resonance in a system of coupled oscillators

We consider a resonantly perturbed system of coupled nonlinear oscillators with small dissipation and outer periodic perturbation. We show that for the large time t～?(-2) one component of the system is described for the most part by the inhomogeneous Mathieu equation while the other component represents pulsation of large amplitude. A Hamiltonian system is obtained which describes for the most part the behavior of the envelope in a special case. The analytic results agree with numerical simulations.     

Non-linear vibrations of a harmonically excited autoparametric system

Harmonic forced vibration of a spring-mass-damper system with a parametrically excited pendulum hinged to the mass is investigated. Two types of restoring forces on the pendulum are considered. The method of harmonic balance is used to evaluate the system response. The results are also verified by numerical integration. Non-periodic system responses are possible if the excitation parameter is large. The performance of the pendulum as an absorber is also studied.     

Phase-locking effects in a system of nonlinear oscillators

Models for the excitation of collective modes in a system of nonlinear classical oscillators, initially out of phase, are discussed. The oscillators may be coupled in a dissipative or conservative manner. The analysis is based on the results of recent studies dealing with the problem of the free excitation of a coherent pulse, analogous to "superradiance" in two-level quantum systems. Several physical examples from the realms of electrodynamics and acoustics are discussed. The processes discussed here may be thought of as chaos-order transitions, provided that "chaos" is understood not as a stochastic nature of an individual oscillator but as the absence of a coherent component in their collective field.     

On the solutions of the coupled system of oscillators and electromagnetic fields excited by an external charge

A systematic study of the longitudinal, transverse and total solutions of a coupled system of oscillators and electromagnetic fields in the presence of an external point charge is carried out. The space-time dependence of the solutions as well as their values in specific cases and asymptotic behavior are analyzed. It is shown that, in general, the longitudinal fields show two well-defined contributions: (a) a symmetric field surrounding the particle and carried convectively which is interpreted as a screening field. (b) an excitation defined in principle in a whole semispace and identified with an oscillator plasma wave which corresponds to the excitation predicted in A. Bohr's microscopic theory of energy losses, although showing somewhat different properties. The transverse solutions appear as differences between the fields given in Fermi's macroscopic theory of energy losses and the longitudinal solutions. Using methods of complex variable theory it is shown how we can separate the total perturbations created by the particle in a medium represented by oscillators into three intimately related contributions: screening, oscillator-plasma excitation and Cherenkov radiation. The space-time configuration of these fields as well as their relation to the longitudinal solutions and their evolution for different ranges of the velocity of the particle is given. The problem of the energy loss associated to the creation of the plasma wave is treated.     

Experimental investigation of the response of a harmonically excited hard Duffing oscillator

A single degree-of-freedom torsional vibratory system, which constitutes a third-order dissipative dynamical system, has been fabricated as a mechanical analogue of hard Duffing equation with strong nonlinearity. The forced response of the system reveals complicated and chaotic motion at low frequency regime. Besides usual jump phenomenon, unpredictable jump phenomenon with two and three coexisting periodic attractors is also observed.      

On wave solutions of a weakly nonlinear and weakly dispersive two-mode wave system

Abstract

In this paper, we introduce and study rigorously a Hamiltonian structure and soliton solutions for a weakly dissipative and weakly nonlinear medium that governs two Korteweg–de vries (KdV) wave modes. The bounded solution and traveling wave solution such as cnoidal wave and solitary wave are obtained. Subsequently, the equation is numerically solved by Fourier spectral method for its two-soliton solution. These solutions may be useful to explain the nonlinear dynamics of waves for an equation supporting multi-mode weakly dispersive and nonlinear wave medium. In addition, we give an explicit explanation of the mathematics behind the soliton phenomenon for a better understanding of the equation.     

Transition from quasi-periodicity to chaos in a system of coupled nonlinear oscillators

The global bifurcation structure for a model of coupled nonlinear oscillators has been analysed numerically. It is shown that destruction of the two-torus preceding chaos is usually observed in this system. The critical surface of the invariant two-torus and its collapse in the course of rotation are firstly observed in a realistic differential equation system. A scaling property for the fine structure of phase-locking regions has also been confirmed.     

Dynamics of a pair of coupled nonlinear oscillators

A pair of coupled classical oscillators with a general potential and general form of coupling is investigated. For general potentials, the single-frequency solution is shown to be stable for small excitations. For special potentials, such system remains stable for an arbitrary excitation. In both cases, the stability does not depend on the form of coupling. Transition to the instability regime follows from the way how nonlinear potential entrains the energy transfer between the oscillators. Relation between the existence of multi-frequency quasi-periodic or periodic solutions and the instability of single-frequency ones is discussed.     

On the use of two classical series expansion methods to determine the vibration of harmonically excited pure cubic oscillators

An analytical approach to determine the steady-state response of a damped and undamped harmonically excited oscillator with no linear term and with cubic non-linearity is presented. The governing equation is transformed into a form suitable for the application of a classical series expansion technique. The Linstedt–Poincaré method and the method of multiple scales are then used to determine the amplitude-frequency response and approximate solution for the response at the excitation frequency. The results obtained are compared with numerical solutions and analytical solutions found in the literature for the case when there is strong non-linearity.     

Transition to coherence in populations of coupled chaotic oscillators: a linear response approach

We consider the collective dynamics in an ensemble of globally coupled chaotic maps. The transition to the coherent state with a macroscopic mean field is analyzed in the framework of the linear response theory. The linear response function for the chaotic system is obtained using the perturbation approach to the Frobenius-Perron operator. The transition point is defined from this function by virtue of the self-excitation condition for the feedback loop. Analytical results for the coupled Bernoulli maps are confirmed by the numerics.     

The window Josephson junction: a coupled linear nonlinear system

We investigate the interface coupling between the two-dimensional sine-Gordon equation and the two-dimensional wave equation in the context of a Josephson window junction using a finite volume numerical method and soliton perturbation theory. The geometry of the domain as well as the electrical coupling parameters are considered. When the linear region is located at each end of the nonlinear domain, we derive an effective one-dimensional model, and using soliton perturbation theory, compute the fixed points that can trap either a kink or antikink at an interface between two sine-Gordon media. This approximate analysis is validated by comparing with the solution of the partial differential equation and describes kink motion in the one-dimensional window junction. Using this, we analyze steady-state kink motion and derive values for the average speed in the one- and two-dimensional systems. Finally, we show how geometry and the coupling parameters can destabilize kink motion.     

Bounds on the trajectories of a system of weakly coupled rotators

In this paper we study a classical mechanical system of weakly coupled rotators on a one-dimensional lattice. Such systems are of interest in statistical mechanics. We prove that for any site in the system there is a large set of initial conditions for which there exists a canonical change of variables such that the trajectory of that site, in the transformed system, is essentially indistinguishable from that of an integrable system for a long (but finite) time. Alternatively, the trajectory of this site lies very close to a torus in the phase space of the system for times very long in comparison with the typical period of the unperturbed rotators. All the estimates in this theory areindependent of the number of degrees of freedom in the system. We propose this mechanism as an explanation of certain numerical experiments.Supported in part by NSF Grant DMS-8403664     

Solvent effects on the linear and nonlinear optical response of silver nanoparticles

Linear and nonlinear (NL) optical properties of silver colloids stabilized with poly(N-vinylpyrrolidone) (PVP) in water, acetone, methanol, and ethylene glycol were studied. Images obtained by transmission electron microscopy reveal narrow size distributions of silver nanoparticles (NPs) with diameters centered at ≈ 6.3 nm (aqueous colloid) and in the 4.3–4.9 nm range for the other colloids. The behavior of the surface plasmon resonance band associated with the NPs was monitored through the linear absorption spectrum, and its dependence on the linear refraction index and the electric dipole moment (EDM) of the solvent molecules was analyzed. The phenomenological parameter, A, obtained from the linear absorption spectra, includes contributions due to the surface effects and the solvent. The third order susceptibility of the colloid was measured using the Z-scan technique at 532 nm, and the NL optical susceptibility of the NPs was determined using the Maxwell–Garnett model. The results indicate that the NL response of the colloids is largely influenced by the molecules adsorbed on the NPs surfaces and the EDM of the solvent molecules. PACS  42.65.-k; 42.65.An; 73.20.Mf; 78.67.-n; 78.67.Bf; 82.70.Dd