Response of a strongly non-linear oscillator to narrowband random excitations

The principal resonance of a van der Pol-Duffing oscillator subject to narrowband random excitations has been studied. By introducing a new expansion parameter the method of multiple scales is adapted for the strongly non-linear system. The behavior of steady state responses, together with their stability, and the effects of system damping and the detuning, and magnitude of the random excitation on steady state responses are analyzed in detail. Theoretical analyses are verified by some numerical results. It is found that when the random noise intensity increases, the steady state solution may change form a limit cycle to a diffused limit cycle, and the system may have two different stable steady state solutions for the same excitation under certain conditions. The results obtained for the strongly non-linear oscillator complement previous results in the literature for weakly non-linear systems.     

线性振子过阻尼和临界阻尼特性对比研究

从理论和实例上对线性振子过阻尼和临界阻尼特性曲线进行分析对比,发现在某特定条件下过阻尼能较快地恢复到平衡位置,还发现从平衡位置开始的线性振子,过阻尼状态时峰值较小,恢复时间较长;而临界阻尼状态下恢复时间较短,峰值较大.     

具有非线性阻尼涨落的线性谐振子的随机共振

较之于线性噪声, 非线性噪声更广泛地存在于实际系统中, 但其研究远不能满足实际情况的需要. 针对作为非线性阻尼涨落噪声基本构成成分的二次阻尼涨落噪声, 本文考虑了周期信号与之共同作用下的线性谐振子, 关注这类具有基本意义的阻尼涨落噪声的非线性对系统共振行为的影响. 利用Shapiro-Loginov公式和Laplace变换推导了系统稳态响应振幅的解析表达式, 并分析了稳态响应振幅的共振行为, 且以数值仿真验证了理论分析的有效性. 研究发现: 系统稳态响应振幅关于非线性阻尼涨落噪声系数具有非单调依赖关系, 特别是非线性阻尼涨落噪声比线性阻尼涨落噪声更有助于增强系统对外部周期信号的响应程度; 而且, 非线性阻尼涨落噪声比线性阻尼涨落噪声使得稳态响应振幅关于噪声强度具有更为丰富的共振行为; 同时, 二次阻尼涨落噪声使得稳态响应振幅关于系统频率出现真正的共振现象; 而在这些现象和性质中, 非线性噪声项的非线性性质对共振行为起着关键的作用. 显然, 以二次阻尼涨落作为基本形式引入的非线性阻尼涨落噪声, 可以有助于提高微弱周期信号检测的灵敏度和实现对周期信号的频率估计.     

A first-principles molecular dynamics approach for predicting optical phonon lifetimes and far-infrared reflectance of polar materials

The Lorentz oscillator model is well-known for its effectiveness to describe the far-infrared optical properties of polar materials. The oscillator strength and damping factor in this model are usually obtained by fitting to experimental data. In this work, a method based on first-principles simulations is developed to parameterize the Lorentz oscillator model without any fitting parameters. The high frequency dielectric constant is obtained from density functional perturbation theory, while the optical phonon frequencies and damping factors are calculated using an analysis of ab initio molecular dynamics trajectories. This method is then used to predict the far-infrared properties of GaAs, and the results are in good agreement with experimental data.     

Fluctuation-induced transitions of a bistable driven polariton system in the presence of damping

Fluctuation-induced transitions from the lower energy state of a bistable nonlinear driven microcavity oscillator are analyzed beyond the Fluctuation–Dissipation theorem. The sources of noise are: both fluctuations of the external pumping and inevitable interaction with the exciton reservoir in the cavity. We show that finite polariton lifetime strongly modifies the phase portrait and influences the temporal parameters of the transition within the bistable regime. To investigate the transient dynamics of the driven polariton system, three different approaches are realized: numerical experiment (1), i.e. direct solution of the quasiclassical dynamic equation for polariton amplitude driven by an external pump, (2) solution of two dimensional Fokker–Planck-equation and (3) effective one dimensional Fokker–Planck-equation, obtained within a low-damping approximation. We show that the escape times obtained within the numerical experiment and the 2D-Fokker–Planck-equation coincide. In contrast, the one dimensional Fokker–Planck-approximation fails for large damping parameters due to strong deviation of the phase trajectories from those obtained within the low-damping approximation. The range of the ratio between damping and detuning for which the 1d approximation is valid, is shown to be smaller than 0.04. Finally, we determine the impact of the fluctuations on experiments illustrated by the narrowing of the hysteresis cycle.     

New Insights into Uniformly Accelerated Detector in a Quantum Field

We obtained an exact solution for a uniformly accelerated Unruh–DeWitt detector interacting with a massless scalar field in (3 + 1) dimensions which enables us to study the entire evolution of the total system, from the initial transient to late-time steady state. We find that the conventional transition probability of the detector from its initial ground state to excited states, as derived from time-dependent perturbation theory over an infinitely long duration of interaction, is valid only in the transient stage and is invalid for cases with proper acceleration smaller than the damping constant. We also found that, unlike in (1 + 1)D results, the (3 + 1)D uniformly accelerated Unruh– DeWitt detector in a steady state does emit a positive radiated power of quantum nature at late-times, but it is not connected to the thermal radiance experienced by the detector in the Unruh effect proper.     

Optimal Runge–Kutta smoothers for the p-multigrid discontinuous Galerkin solution of the 1D Euler equations

This work presents a family of original Runge–Kutta methods specifically designed to be effective relaxation schemes in the numerical solution of the steady state solution of purely advective problems with a high-order accurate discontinuous Galerkin space discretization and a p-multigrid solution algorithm. The design criterion for the construction of the Runge–Kutta methods here developed is different form the one traditionally used to derive optimal Runge–Kutta smoothers for the h-multigrid algorithm, which are designed in order to provide a uniform damping of the error modes in the high-frequency range only. The method here proposed is instead designed in order to provide a variable amount of damping of the error modes over the entire frequency spectrum. The performance of the proposed schemes is assessed in the solution of the steady state quasi one-dimensional Euler equations for two test cases of increasing difficulty. Some preliminary results showing the performance for multidimensional applications are also presented.     

Pure odd-order oscillators with constant excitation

In this paper the excited vibrations of a truly nonlinear oscillator are analyzed. The excitation is assumed to be constant and the nonlinearity is pure (without a linear term). The mathematical model is a second-order nonhomogeneous differential equation with strong nonlinear term. Using the first integral, the exact value of period of vibration i.e., angular frequency of oscillator described with a pure nonlinear differential equation with constant excitation is analytically obtained. The closed form solution has the form of gamma function. The period of vibration depends on the value of excitation and of the order and coefficient of the nonlinear term. For the case of pure odd-order-oscillators the approximate solution of differential equation is obtained in the form of trigonometric function. The solution is based on the exact value of period of vibration. For the case when additional small perturbation of the pure oscillator acts, the so called ‘Cveticanin's averaging method’ for a truly nonlinear oscillator is applied. Two special cases are considered: one, when the additional term is a function of distance, and the second, when damping acts. To prove the correctness of the method the obtained results are compared with those for the linear oscillator. Example of pure cubic oscillator with constant excitation and linear damping is widely discussed. Comparing the analytically obtained results with exact numerical ones it is concluded that they are in a good agreement. The investigations reported in the paper are of special interest for those who are dealing with the problem of vibration reduction in the oscillator with constant excitation and pure nonlinear restoring force the examples of which can be found in various scientific and engineering systems. For example, such mechanical systems are seats in vehicles, supports for machines, cutting machines with periodical motion of the cutting tools, presses, etc. The examples can be find in electronics (electromechanical devices like micro-actuators and micro oscillators), in music instruments (hammers in piano), in human voice producing folds (voice cords), etc.     

The dynamics of two coupled Van der Pol oscillators with attractive and repulsive coupling

We consider a variant of two coupled Van der Pol oscillators with both attractive and repulsive mean-field interactions. In the presence of attractive coupling, the system is in the complete synchrony, while repulsive coupling shows anti-synchronization state leading to suppression of oscillations with increasing interaction strength. The coupled system with both attractive and repulsive interactions shows competitive tendencies of being complete synchronization and anti-synchronization resulting in the stabilization of the fixed point. We have also studied the effect of the damping coefficient of the VdP oscillator on the nature of the transition from oscillatory to a steady-state. These oscillators stabilize to unstable equilibrium point or coupling dependent inhomogeneous steady state via second or first-order transitions respectively depending upon the damping coefficient and coupling strength. These transitions are analyzed in the parameter plane by analytical and numerical studies of the two coupled Van der Pol oscillators.     

混沌振子系统周期解几何特征量分析与微弱周期信号的定量检测

提出了一种对微弱周期信号的定量检测方法.分析混沌振子系统在大尺度周期状态下的相对稳定输出时,发现了混沌振子系统输出周期解的平均面积是一个比较稳定的几何特征量.该几何特征量与待测信号幅值之间存在比较稳定的单调递增关系.在一定的参数条件下,几何特征量精度可达到10^-6V².利用混沌系统对随机噪声信号的免疫性和对微弱周期信号的敏感性,进一步建立了微弱周期信号的定量检测方法.仿真实验表明,随着待检测幅度的增加,在保证检测精度的同时,抗噪性能也随之增强. 关键词： 混沌振子系统 大尺度周期相态 周期解的几何特征量 微弱周期信号的定量检测     

谐和与随机噪声联合作用Duffing单边约束系统的响应

研究了Duffing单边约束系统在谐和与随机噪声联合激励下的响应问题. 用谐波平衡法和摄动法分析了系统在确定性谐和激励和随机激励联合作用下的响应,用随机平均法讨论了随机扰动项对系统响应的影响. 在一定条件下,当约束距离较大时对应于不同的初始条件,系统具有两个非碰撞的稳态响应;而当约束距离不大时,对应于不同的初始条件,系统也可以有两个不同的稳态响应,其中一个是发生碰撞的响应,而另外一个则不发生碰撞. 数值模拟表明该方法是有效的. 关键词： Duffing单边约束系统 随机响应 谐波平衡法 摄动法     

The general anharmonicx 2q+2 damped oscillator and the coherent state representation

We obtain here a perturbative solution of the generalx ^2q+2 anharmonic damped oscillator in the coherent state representation. The solution does not contain any secular term and shows, explicitly, the damping and the anharmonic effects.     

Time correlation functions and spectral distributions of the duffing oscillator in a random force field

The Brownian motion of the Duffing oscillator is analyzed in the case when the oscillator damping is small compared with its frequency, whereas the nonlinearity may be arbitrary. The expressions for the time-correlation functions of coordinates are obtained in an explicit form. If the nonlinearity is small, the dynamics of the system is shown to be determined by a relation between the frequency straggling due to fluctuations of the amplitude and damping. At large nonlinearity the correlators do not depend on the damping. The frequency dependences of the spectral representations of the correlators of coordinates are investigated for various ratios between the oscillator parameters.     

Response of SDOF nonlinear oscillators with lightly fractional derivative damping under real noise excitations

This paper deals with the response of single-degree-of-freedom (SDOF) strongly nonlinear oscillator with lightly fractional derivative damping to external and (or) parametric real noise excitations. First, the state vector of the displacement and the velocity is approximated by one-dimensional time-homogeneous diffusive Markov process of amplitude through using the stochastic averaging method. Then, the stationary probability density of amplitude is obtained by solving the Fokker-Planck-Kolmogorov (FPK) equation associated with the averaged It? equation of amplitude, in which the Fourier series expansions are used to obtain the explicit expressions of the drift and diffusion coefficients. Finally, the response of a Duffing oscillator with lightly fractional derivative damping under external and parametric real noise excitations is evaluated by using the proposed procedure and compared with that from the Monte Carlo simulation of original system.     

Damping in metals under combined stress loading

Studies have been made of damping behaviour under conditions of combined stress (i.e., biaxial loading) for six high damping metals—two alloys of manganese-copper and four grades of cast iron. Measured values of damping are presented from tests covering a range of combined stress states and the damping behaviour is interpreted in terms of dependence on stress state. The measured values were obtained from a new experiment in which various combined stress conditions are generated by coupled torsion/bending vibration of a cantilever. The range covered is from all torsion to all bending, i.e., principal stress ratios from ?1 to 0. Damping is determined from energy input during steady state resonant vibration, but frequency response and free decay methods can also be used. Details of the apparatus are given.     

周期场作用下自旋的主方程及其稳定态

用Robertson理论推导一个在纵磁场和横周期场作用下的自旋的运动方程。得到了阻尼和自旋稳定态对外场强度、频率及与热库的耦合系数的依赖关系。讨论了自旋稳定态的结构及其形成原因。 关键词：     

Vibronic Sidebands of Coherent Phonon States

Recently experiments have been reported about phonon sidebands in doped crystals, which may originate from coherent phonon states. The corresponding modes are either confined phonon modes in nanocrystals or localized phonon modes in bulk materials, both showing small damping due to phonon-phonon interaction. We present a theory of the lineshape of vibronic sideband spectra due to coherent phonon states using the conventional model of linear electron-phonon coupling and displaced equilibrium positions of the oscillators in the initial and final electronic states. Unlike in the conventional theory, the initial state of the oscillator is taken as a coherent phonon state and not as a thermalized one. Under these conditions we got an exact analytical solution for the lineshape of the vibronic sideband. The lineshape is determined by two parameters, the Huang-Rhys parameter S and the coherence parameter α of the phonon state. For α = 0 the lineshape converts into the standard Pekarian form for T = 0.     

The formation of correlated states and the increase in barrier transparency at a low particle energy in nonstationary systems with damping and fluctuations

We consider peculiarities in the formation of a coherent correlated state (CCS) of a particle in a periodically modulated harmonic oscillator with damping for various types of stochastic perturbation. It is shown that in the absence of stochastic perturbation, an optimal relation exists between the damping parameter (damping coefficient) and the modulation depth, for which the ??extrinsic?? characteristics of the oscillator (amplitudes of ??classical?? oscillation and the momentum of a particle) remain unchanged, while the correlation coefficient rapidly increases from |r| = 0 to |r|_max ?? 1; this corresponds to a completely correlated coherent state. Under nonoptimal conditions, the formation of the CCS with a simultaneous increase in is accompanied by damping or excitation of the oscillator. It is shown that for a certain relation between the damping coefficient and the modulation depth, the presence of a stochastic external force acting on the nonstationary oscillator does not prevent the formation of a CCS with |r|_max ?? 1. A fundamentally different effect is observed under a stochastic influence on the nonstationary frequency of the oscillator; this effect always limits the value of |r| at a level |r|_max < 1; a CCR cannot be formed with an unlimited increase in its intensity, and |r|_max ?? 0. The influence of the CCS formation on the averaged probability ??D?? of the tunnel effect (transparency of the potential barrier) is considered for a particle in an oscillator with damping both in the absence and in the presence of a stochastic force. It is shown using a specific example that complete clearing of the potential barrier and the increase in the barrier transparency from the initial value ??D _r=0?? = 10^?80 to ??D?? ?? 1 can occur over a comparatively short time interval in both these cases. These effects can be used to obtain highly efficient nuclear fusion at a low energy of interacting particles.     

Analysis of modified Van der Pol’s oscillator using He’s parameter-expanding methods

In this work, a powerful analytical method, called He’s parameter-expanding methods (HPEM) is used to obtain the exact solutions of non-linear modified Van der Pol’s oscillator. The classical Van der Pol equation with delayed feedback and a modified equation where a delayed term provides the damping are considered. It is shown that one term in series expansions is sufficient to obtain a highly accurate solution, which is valid for the whole solution domain. Comparison of the obtained solution with those obtained using perturbation method shows that this method is effective and convenient to solve this problem. This method introduces a capable tool to solve this kind of non-linear problems.     

A general mass law for broadband energy harvesting

It is well known that the power absorbed by a linear oscillator when excited by white noise base acceleration depends only on the mass of the oscillator and the spectral density of the base motion. This places an upper bound on the energy that can be harvested from a linear oscillator under broadband excitation, regardless of the stiffness of the system or the damping factor. It is shown here that the same result applies to any multi-degree-of-freedom nonlinear system that is subjected to white noise base acceleration: for a given spectral density of base motion the total power absorbed is proportional to the total mass of the system. The only restriction to this result is that the internal forces are assumed to be a function of the instantaneous value of the state vector. The result is derived analytically by several different approaches, and numerical results are presented for an example two-degree-of-freedom-system with various combinations of linear and nonlinear damping and stiffness.