An energy-based computational method in the analysis of the transmission of energy in a chain of coupled oscillators

In this paper we study the phenomenon of nonlinear supratransmission in a semi-infinite discrete chain of coupled oscillators described by modified sine-Gordon equations with constant external and internal damping, and subject to harmonic external driving at the end. We develop a consistent and conditionally stable finite-difference scheme in order to analyze the effect of damping in the amount of energy injected in the chain of oscillators; numerical bifurcation analyses to determine the dependence of the amplitude at which supratransmission first occurs with respect to the frequency of the driving oscillator are carried out in order to show the consequences of damping on harmonic phonon quenching and the delay of appearance of critical amplitude

A Monte Carlo Method for Modeling Thermal Damping: Beyond the Brownian-Motion Master Equation

The "standard" Brownian motion master equation, used to describe thermal damping, is not completely positive, and does not admit a Monte Carlo method, important in numerical simulations. To eliminate both these problems one must add a term that generates additional position diffusion. He we show that one can obtain a completely positive simple quantum Brownian motion, efficiently solvable, without any extra diffusion. This is achieved by using a stochastic Schroedinger equation (SSE), closely analogous to Langevin's equation, that has no equivalent Markovian master equation. Considering a specific example, we show that this SSE is sensitive to nonlinearities in situations in which the master equation is not, and may therefore be a better model of damping for nonlinear systems.Comment: 6 pages, revtex4. v2: numerical results for a nonlinear syste

Frequency domain energy transfer properties of bilinear oscillators under harmonic loadings

In this paper, the energy transfer phenomenon of bilinear oscillators in the frequency domain is analyzed using the new concept of Nonlinear Output Frequency Response Functions (NOFRFs). The analysis provides insight into how new frequency generation can occur using bilinear oscillators, and reveals, for the first time, that it is the resonant frequencies of the NOFRFs that dominate the occurrence of this well-known nonlinear behaviour. The results are of significance for the design and fault diagnosis of mechanical systems and structures which can be described by a bilinear oscillator model

Analysis of bilinear oscillators under harmonic loading using nonlinear output frequency response functions

In this paper, the new concept of Nonlinear Output Frequency Response Functions (NOFRFs) is extended to the harmonic input case, an input-independent relationship is found between the NOFRFs and the Generalized Frequency Response Functions (GFRFs). This relationship can greatly simplify the application of the NOFRFs. Then, beginning with the demonstration that a bilinear oscillator can be approximated using a polynomial type nonlinear oscillator, the NOFRFs are used to analyze the energy transfer phenomenon of bilinear oscillators in the frequency domain. The analysis provides insight into how new frequency generation can occur using bilinear oscillators and how the sub-resonances occur for the bilinear oscillators, and reveals that it is the resonant frequencies of the NOFRFs that dominate the occurrence of this well-known nonlinear behaviour. The results are of significance for the design and fault diagnosis of mechanical systems and structures which can be described by a bilinear oscillator model

Harvesting Thermal Fluctuations: Activation Process Induced by a Nonlinear Chain in Thermal Equilibrium

We present a model in which the immediate environment of a bistable system is a molecular chain which in turn is connected to a thermal environment of the Langevin form. The molecular chain consists of masses connected by harmonic or by anharmonic springs. The distribution, intensity, and mobility of thermal fluctuations in these chains is strongly dependent on the nature of the springs and leads to different transition dynamics for the activated process. Thus, all else (temperature, damping, coupling parameters between the chain and the bistable system) being the same, the hard chain may provide an environment described as diffusion-limited and more effective in the activation process, while the soft chain may provide an environment described as energy-limited and less effective. The importance of a detailed understanding of the thermal environment toward the understanding of the activation process itself is thus highlighted

Optically mediated nonlinear quantum optomechanics

We consider theoretically the optomechanical interaction of several mechanical modes with a single quantized cavity field mode for linear and quadratic coupling. We focus specifically on situations where the optical dissipation is the dominant source of damping, in which case the optical field can be adiabatically eliminated, resulting in effective multimode interactions between the mechanical modes. In the case of linear coupling, the coherent contribution to the interaction can be exploited e.g. in quantum state swapping protocols, while the incoherent part leads to significant modifications of cold damping or amplification from the single-mode situation. Quadratic coupling can result in a wealth of possible effective interactions including the analogs of second-harmonic generation and four-wave mixing in nonlinear optics, with specific forms depending sensitively on the sign of the coupling. The cavity-mediated mechanical interaction of two modes is investigated in two limiting cases, the resolved sideband and the Doppler regime. As an illustrative application of the formal analysis we discuss in some detail a two-mode system where a Bose-Einstein condensate is optomechanically linearly coupled to the moving end mirror of a Fabry-P\'erot cavity.Comment: 11 pages, 8 figure

Effects of noise on the internal resonance of a nonlinear oscillator

We numerically analyze the response to noise of a system formed by two coupled mechanical oscillators, one of them having Duffing and van der Pol nonlinearities, and being excited by a self-sustaining force proportional to its own velocity. This system models the internal resonance of two oscillation modes in a vibrating solid beam clamped at both ends. In applications to nano- and micromechanical devices, clamped-clamped beams are subjected to relatively large thermal and electronic noise, so that characterizing the fluctuations induced by these effects is an issue of both scientific and technological interest. We pay particular attention to the action of stochastic forces on the stability of internal-resonance motion, showing that resonant oscillations become more robust than other forms of periodic motion as the quality factor of the resonant mode increases. The dependence on other model parameters - in particular, on the coupling strength between the two oscillators - is also assessed.Fil: Zanette, Damian Horacio. Comisión Nacional de Energía Atómica. Gerencia del Area de Investigación y Aplicaciones No Nucleares. Gerencia de Física (Centro Atómico Bariloche); Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

It takes a village to raise a tide: nonlinear multiple-mode coupling and mode identification in KOI-54

We explore the tidal excitation of stellar modes in binary systems using Kepler observations of the remarkable eccentric binary KOI-54 (HD 187091; KIC 8112039), which displays strong ellipsoidal variation as well as a variety of linear and nonlinear pulsations. We report the amplitude and phase of over 120 harmonic and anharmonic pulsations in the system. We use pulsation phases to determine that the two largest-amplitude pulsations, the 90th and 91st harmonics, most likely correspond to axisymmetric m=0 modes in both stars, and thus cannot be responsible for resonance locks as had been recently proposed. We find evidence that the amplitude of at least one of these two pulsations is decreasing with a characteristic timescale of ~100 yr. We also use the pulsations' phases to confirm the onset of the traveling wave regime for harmonic pulsations with frequencies <~50 Omega_orbit, in agreement with theoretical expectations. We present evidence that many pulsations that are not harmonics of the orbital frequency correspond to modes undergoing simultaneous nonlinear coupling to multiple linearly driven parent modes. Since coupling among multiple modes can lower the threshold for nonlinear interactions, nonlinear phenomena may be easier to observe in highly eccentric systems, where broader arrays of driving frequencies are available. This may help to explain why the observed amplitudes of the linear pulsations are much smaller than the theoretical threshold for decay via three-mode coupling.Comment: Accepted for publication in MNRAS. Only minor corrections. 16 Pages; 8 Figures; 3 Table