Periodic motions and resonances of impact oscillators

Bilinear oscillators – the oscillators whose springs have different stiffnesses in compression and tension – model a wide range of phenomena. A limiting case of bilinear oscillator with infinite stiffness in compression – the impact oscillator – is studied here. We investigate a special set of impact times – the eigenset, which corresponds to the solution of the homogeneous equation, i.e. the oscillator without the driving force. We found that this set and its subsets are stable with respect to variation of initial conditions. Furthermore, amongst all periodic sets of impact times with the period commensurate with the period of driving force, the eigenset is the only one which can support resonances, in particular the multi-‘harmonic’ resonances. Other resonances should produce non-periodic sets of impact times. This funding indicates that the usual simplifying assumption [e.g., S.W. Shaw, P.J. Holmes, A periodically forced piecewise linear oscillator, Journal of Sound and Vibration 90 (1983) 129–155] that the times between impacts are commensurate with the period of the driving force does not always hold. We showed that for the first sub-‘harmonic resonance’ – the resonance achieved on a half frequency of the main resonance – the set of impact times is asymptotically close to the eigenset. The envelope of the oscillations in this resonance increases as a square root of time, opposite to the linear increase characteristic of multi-‘harmonic’ resonances.     

Synchronization in arrays of coupled self-induced friction oscillators

We investigate synchronization phenomena in systems of self-induced dry friction oscillators with kinematic excitation coupled by linear springs. Friction force is modelled according to exponential model. Initially, a single degree of freedom mass-spring system on a moving belt is considered to check the type of motion of the system (periodic, non-periodic). Then the system is coupled in chain of identical oscillators starting from two, up to four oscillators. A reference probe of two coupled oscillators is applied in order to detect synchronization thresholds for both periodic and non-periodic motion of the system. The master stability function is applied to predict the synchronization thresholds for longer chains of oscillators basing on two oscillator probe. It is shown that synchronization is possible both for three and four coupled oscillators under certain circumstances. Our results confirmed that this technique can be also applied for the systems with discontinuities.     

Landauer formula for phonon heat conduction: relation between energy transmittance and transmission coefficient

A Fibonacci chain is composed of oscillators with two different masses and its total (even) number of oscillators N is associated with Fibonacci numbers. The momentum autocorrelation function of a specific oscillator in the chain is shown to be combination of 1?+?(N/2) cosines, their frequencies and amplitudes are calculated numerically for N????40. The momentum autocorrelation functions are illustrated for Fibonacci chains with N up to 176.     

Four mass coupled oscillator guitar model

Coupled oscillator models have been used for the low frequency response (50 to 250 Hz) of a guitar. These 2 and 3 mass models correctly predict measured resonance frequency relationships under various laboratory boundary conditions, but did not always represent the true state of a guitar in the players' hands. The model presented has improved these models in three ways, (1) a fourth oscillator includes the guitar body, (2) plate stiffnesses and other fundamental parameters were measured directly and effective areas and masses used to calculate the responses, including resonances and phases, directly, and (3) one of the three resultant resonances varies with neck and side mass and can also be modeled as a bar mode of the neck and body. The calculated and measured resonances and phases agree reasonably well.     

Predicting optimal drive sweep rates for autoresonance in Duffing-type oscillators: A beat method using Teager-Kaiser instantaneous frequency

Sustained resonance in a linear oscillator is achievable with a drive whose constant frequency matches the resonant frequency of the oscillator. But in oscillators with nonlinear restoring forces such as the pendulum, Duffing and Duffing-Van der Pol oscillator, the resonant frequency changes as the amplitude changes, so a constant frequency drive results in a beat oscillation instead of sustained resonance. Duffing-type nonlinear oscillators can be driven into sustained resonance, called autoresonance, when the drive frequency is swept in time to match the changing resonant frequency of the oscillator. We find that near-optimal drive linear sweep rates for autoresonance can be estimated from the beat oscillation resulting from constant frequency excitation. Specifically, a least squares estimate of the Teager-Kaiser instantaneous frequency versus time for the beat response to a stationary drive provides a near-optimal estimate of the nonstationary drive linear sweep rate needed to sustain resonance in the pendulum, Duffing and Duffing-Van der Pol oscillators. We confirm these predictions with model-based numerical simulations. An advantage of the beat method of estimating optimal drive sweep rates for maximal autoresonant response is that no model is required so experimentally generated beat oscillation data can be used for systems where no model is available.     

Justification of the “symmetric damping” model of the dynamical Casimir effect in a cavity with a semiconductor mirror*

A “microscopic” justification of the “symmetric damping” model of a quantum oscillator with time-dependent frequency and time-dependent damping is given. This model is used to predict the results of experiments on simulating the dynamical Casimir effect in a cavity with a photo-excited semiconductor mirror. It is shown that the most general bilinear time-dependent coupling of a selected oscillator (field mode) to a bath of harmonic oscillators results in two equal friction coefficients for the both quadratures, provided all the coupling coefficients are proportional to a single arbitrary function of time whose duration is much shorter than the periods of all oscillators. The choice of coupling in the rotating wave approximation form leads to the “minimum noise” model of the quantum damped oscillator, introduced earlier in a pure phenomenological way.     

Resonance tongues in a system of globally coupled FitzHugh-Nagumo oscillators with time-periodic coupling strength

We investigate the dynamics of a population of globally coupled FitzHugh-Nagumo oscillators with a time-periodic coupling strength. While for synchronizing global coupling, the in-phase state is always stable, the oscillators split into several cluster states for desynchronizing global coupling, most commonly in two, irrespective of the coupling strength. This confines the ability of the system to form n:m locked states considerably. The prevalence of two and four cluster states leads to large 2:1 and 4:1 subharmonic resonance regions, while at low coupling strength for a harmonic 1:1 or a superharmonic 1:m time-periodic coupling coefficient, any resonances are absent and the system exhibits nonresonant phase drifting cluster states. Furthermore, in the unforced, globally coupled system the frequency of the oscillators in a cluster state is in general lower than that of the uncoupled oscillator and strongly depends on the coupling strength. Periodic variation of the coupling strength at twice the natural frequency causes each oscillator to keep oscillating with its autonomous oscillation period.     

Simulation of thermoelastic properties of solids in the framework of an ensemble of anharmonic oscillators

It has been shown that, in a classical ensemble of anharmonic oscillators, the mean value of the oscillator coordinate is a classical parameter in the sense that the statistical sum of the ensemble satisfies, to the second order in the anharmonicity constant, the stationary condition with respect to this parameter. This stationary condition is equivalent to the classical condition for the balance of external and internal forces acting on the oscillator. This equivalence is justified by the fact that the statistical sum, which is stationary with respect to the mean oscillator coordinate, agrees within this accuracy with the usual statistical sum of independent anharmonic oscillators. After introducing the classical parameter into a large thermodynamic system, the energy balance under the mechanical deformation of the system is realized through the exchange between two scale levels: the energy of oscillations at the microlevel and the macroscopic potential energy of deformation of the sample as a whole.     

A time-dependent density functional theory investigation of plasmon resonances of linear Au atomic chains

We report theoretical studies on the plasmon resonances in linear Au atomic chains by using ab initio time-dependent density functional theory. The dipole responses are investigated each as a function of chain length. They converge into a single resonance in the longitudinal mode but split into two transverse modes. As the chain length increases, the longitudinal plasmon mode is redshifted in energy while the transverse modes shift in the opposite direction (blueshifts). In addition, the energy gap between the two transverse modes reduces with chain length increasing. We find that there are unique characteristics, different from those of other metallic chains. These characteristics are crucial to atomic-scale engineering of single-molecule sensing, optical spectroscopy, and so on.     

Chains of oscillators with negative stiffness elements

Negative stiffness is not allowed by thermodynamics and hence materials and systems whose global behaviour exhibits negative stiffness are unstable. However the stability is possible when these materials/systems are elements of a larger system sufficiently stiff to stabilise the negative stiffness elements. In order to investigate the effect of stabilisation we analyse oscillations in a chain of n linear oscillators (masses and springs connected in series) when some of the springs? stiffnesses can assume negative values. The ends of the chain are fixed. We formulated the necessary stability condition: only one spring in the chain can have negative stiffness. Furthermore, the value of negative stiffness cannot exceed a certain critical value that depends upon the (positive) stiffnesses of other springs. At the critical negative stiffness the system develops an eigenmode with vanishing frequency. In systems with viscous damping vanishing of an eigenfrequency does not yet lead to instability. Further increase in the value of negative stiffness leads to the appearance of aperiodic eigenmodes even with light damping. At the critical negative stiffness the low dissipative mode becomes non-dissipative, while for the high dissipative mode the damping coefficient becomes as twice as high as the damping coefficient of the system. A special element with controllable negative stiffness is suggested for designing hybrid materials whose stiffness and hence the dynamic behaviour is controlled by the magnitude of applied compressive force.     

Abatement of thermal noise due to internal damping in 2D oscillators with rapidly rotating test masses

Mechanical oscillators can be sensitive to very small forces. Low frequency effects are up-converted to higher frequency by rotating the oscillator. We show that for 2-dimensional oscillators rotating at frequency much higher than the signal the thermal noise force due to internal losses and competing with it is abated as the square root of the rotation frequency. We also show that rotation at frequency much higher than the natural one is possible if the oscillator has 2 degrees of freedom, and describe how this property applies also to torsion balances. In addition, in the 2D oscillator the signal is up-converted above resonance without being attenuated as in the 1D case, thus relaxing requirements on the read out. This work indicates that proof masses weakly coupled in 2D and rapidly rotating can play a major role in very small force physics experiments.     

End and central plasmon resonances in linear atomic chains

The existence and nature of end and central plasmon resonances in a linear atomic chain, the 1D analog to surface and bulk plasmons in 2D metals, has been predicted by ab initio time-dependent density functional theory. Length dependence of the absorption spectra shows the emergence and development of collectivity of these resonances. It converges to a single resonance in the longitudinal mode, and two transverse resonances, which are localized at the ends and center of the atom chains. These collective modes bridge the gaps, in concept and scale, between the collective excitation of atomic physics and nanoplasmonics. It also outlines a route to atomic-scale engineering of collective excitations.     

On the stability properties of a damped oscillator with a periodically time-varying mass

In this paper the vibrations of a damped, linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be considered. Both the free and forced vibrations of the oscillator will be studied. For the free vibrations the minimal damping rates will be computed, for which the oscillator is always stable. The forced vibrations are partly due to small masses, which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Additionally, an external harmonic force will be applied to the oscillator. Not only solutions of the oscillator equations will be constructed, but also stability properties for the free, and for the forced vibrations will be presented for various parameter values. For the external, harmonic forcing case an interesting resonance condition will be derived.     

Dipole emission of a nanostructure system composed of two atoms and dimensional resonances

A stationary solution is obtained for the joint system of equations for atomic and field variables for two different atoms with dipole-dipole interaction in the radiation field taking into account the common radiative friction. The atoms are treated as an Lorentz oscillator with one isolated resonance. The interaction of atoms in the radiation field forms four dimensional resonances at frequencies that are substantially different from the natural frequencies of isolated atoms. Two of the four dimensional resonances are characterized by negative dispersion, and the intensity of dipole emission at these frequencies may be increased with respect to the intensity of emission at the frequencies of natural atomic resonances by a factor of about 10¹².     

Effective long-time phase dynamics of limit-cycle oscillators driven by weak colored noise

An effective white-noise Langevin equation is derived that describes long-time phase dynamics of a limit-cycle oscillator driven by weak stationary colored noise. Effective drift and diffusion coefficients are given in terms of the phase sensitivity of the oscillator and the correlation function of the noise, and are explicitly calculated for oscillators with sinusoidal phase sensitivity functions driven by two typical colored Gaussian processes. The results are verified by numerical simulations using several types of stochastic or chaotic noise. The drift and diffusion coefficients of oscillators driven by chaotic noise exhibit anomalous dependence on the oscillator frequency, reflecting the peculiar power spectrum of the chaotic noise.     

Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics

We continue the study of a model for heat conduction [6] consisting of a chain of non-linear oscillators coupled to two Hamiltonian heat reservoirs at different temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite-dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators. Received: 12 March 2001 / Accepted: 5 August 2001     

High-order subharmonic parametric resonance of nonlinearly coupled micromechanical oscillators

This paper studies parametric resonance of coupled micromechanical oscillators under periodically varying nonlinear coupling forces. Different from most of previous related works in which the periodically varying coupling forces between adjacent oscillators are linearized, our work focuses on new physical phenomena caused by the periodically varying nonlinear coupling. Harmonic balance method (HBM) combined with Newton iteration method is employed to find steady-state periodic solutions. Similar to linearly coupled oscillators studied previously, the present model predicts superharmonic parametric resonance and the lower-order subharmonic parametric resonance. On the other hand, the present analysis shows that periodically varying nonlinear coupling considered in the present model does lead to the appearance of high-order subharmonic parametric resonance when the external excitation frequency is a multiple or nearly a multiple (≥3) of one of the natural frequencies of the oscillator system. This remarkable new phenomenon does not appear in the linearly coupled micromechanical oscillators studied previously, and makes the range of exciting resonance frequencies expanded to infinity. In addition, the effect of a linear damping on parametric resonance is studied in detail, and the conditions for the occurrence of the high-order subharmonics with a linear damping are discussed.     

STOCHASTIC AVERAGING OF STRONGLY NON-LINEAR OSCILLATORS UNDER BOUNDED NOISE EXCITATION

A stochastic averaging method for strongly non-linear oscillators under external and/or parametric excitation of bounded noise is proposed by using the so-called generalized harmonics functions. The method is then applied to study the primary resonance of Duffing oscillator with hardening spring under external excitation of bounded noise. The stochastic jump and its bifurcation of the system are observed and explained by using the stationary probability density of amplitude and phase. Subsequently, the method is applied to study the dynamical instability and parametric resonance of Duffing oscillator with hardening spring under parametric excitation of bounded noise. The primary unstable region is delineated by evaluating the Lyapunov exponent of linearized system, and the response and jump of non-linear system around the unstable region are examined by using the sample functions and stationary probability density of amplitude and phase.     

外力驱动下耦合弹簧振子的法诺共振

法诺共振是物理体系中普遍存在的一种非对称共振现象,它最早起源于量子物理,其微观图像是原子谱线中窄的分离态与宽的连续态之间的相干干涉.本文利用经典力学体系中两个弹簧振子的耦合,使其中一个弹簧振子受到周期性外力的驱动,成功类比了量子力学中的法诺共振现象.通过分析每个弹簧振子的动力学方程,严格求解它们的振动公式,从而得到每个弹簧振子的振幅和位相与外部驱动力频率之间的关系.结果表明,耦合体系中受外力驱动的那个弹簧振子既可以发生非对称的法诺共振,又可以发生对称的洛仑兹共振,而另一个弹簧振子只能发生洛仑兹共振.本文的推导与分析能够使读者更好地理解法诺共振现象及其激发条件.     

Nonlinear dynamics of oscillators with bilinear hysteresis and sinusoidal excitation

The transient and steady-state response of an oscillator with hysteretic restoring force and sinusoidal excitation are investigated. Hysteresis is modeled by using the bilinear model of Caughey with a hybrid system formulation. A novel method for obtaining the exact transient and steady-state response of the system is discussed. Stability and bifurcations of periodic orbits are studied using Poincaré maps. Results are compared with asymptotic expansions obtained by Caughey. The bilinear hysteretic element is found to act like a ‘soft spring’. Several sub-harmonic resonances are found in the system, however, no chaotic behavior is observed. Away from the sub-harmonic resonance the asymptotic expansions and the exact steady-state response of the system are seen to match with good accuracy.