Application of a modified rational harmonic balance method for a class of strongly nonlinear oscillators

An analytical approximate technique for conservative nonlinear oscillators is proposed. This method is a modification of the rational harmonic balance method in which analytical approximate solutions have rational form. This approach gives us the frequency of the motion as a function of the amplitude of oscillation. We find that this method works very well for the whole range of parameters, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed. The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values and the results reveal that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems with complex nonlinearities.     

研究强非线性振动系统同宿分岔问题的规范形方法

以改进的规范形理论为基础,采用强非线性振动问题的分析方法,拓展了原有弱非线性振动系统同宿分岔判据的适用范围.首先在复规范形求解过程中引入待定固有频率,计算了一类单自由度强非线性振动系统的周期解.然后分别依据系统的待定固有频率趋于零和周期轨道趋近于鞍点两条途径获得了强非线性振动条件下系统同宿分岔的解析判据.最后通过与原有解析结果和数值结果相比较验证了本文方法的有效性. 关键词： 规范形 同宿分岔 强非线性 周期解     

Application of He's parameter-expansion method for oscillators with smooth odd nonlinearities

This Letter applies He's parameter-expansion method to oscillators with smooth nonlinearities. The method does not depend upon small parameter assumption, hence it is very better than the perturbation method. In parameter-expansion method the solution and unknown frequency of oscillation are expanded in a series by a bookkeeping parameter. By imposing the non-secularity condition at each order in the expansion the method provides different approximations to both the solution and the frequency of oscillation. One iteration step provides an approximate solution which is valid for the whole solution domain. The method can be easily extended to other nonlinear oscillations.     

Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He's homotopy perturbation method

He's homotopy perturbation method is used to calculate higher-order approximate periodic solutions of a nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(x). We find He's homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 1.56% for all values of oscillation amplitude, while this relative error is 0.30% for the second iteration and as low as 0.057% when the third-order approximation is considered. Comparison of the result obtained using this method with those obtained by different harmonic balance methods reveals that He's homotopy perturbation method is very effective and convenient.     

Dynamic properties of combustion instability in a lean premixed gas-turbine combustor

We experimentally investigate the dynamic behavior of the combustion instability in a lean premixed gas-turbine combustor from the viewpoint of nonlinear dynamics. A nonlinear time series analysis in combination with a surrogate data method clearly reveals that as the equivalence ratio increases, the dynamic behavior of the combustion instability undergoes a significant transition from stochastic fluctuation to periodic oscillation through low-dimensional chaotic oscillation. We also show that a nonlinear forecasting method is useful for predicting the short-term dynamic behavior of the combustion instability in a lean premixed gas-turbine combustor, which has not been addressed in the fields of combustion science and physics.     

Harmonic balancing approach to nonlinear oscillations of a punctual charge in the electric field of charged ring

The harmonic balance method is used to construct approximate frequency-amplitude relations and periodic solutions to an oscillating charge in the electric field of a ring. By combining linearization of the governing equation with the harmonic balance method, we construct analytical approximations to the oscillation frequencies and periodic solutions for the oscillator. To solve the nonlinear differential equation, firstly we make a change of variable and secondly the differential equation is rewritten in a form that does not contain the square-root expression. The approximate frequencies obtained are valid for the complete range of oscillation amplitudes and excellent agreement of the approximate frequencies and periodic solutions with the exact ones are demonstrated and discussed.     

An RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in a Lagrangian coordinate

In this paper,Runge-Kutta Discontinuous Galerkin(RKDG) finite element method is presented to solve the onedimensional inviscid compressible gas dynamic equations in a Lagrangian coordinate.The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method.A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method.For multi-medium fluid simulation,the two cells adjacent to the interface are treated differently from other cells.At first,a linear Riemann solver is applied to calculate the numerical ?ux at the interface.Numerical examples show that there is some oscillation in the vicinity of the interface.Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical ?ux at the interface,which suppresses the oscillation successfully.Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.     

Some aspects of the harmonic balance method applied to the clarinet

The clarinet has been extensively studied by various theoretical and experimental techniques. In this paper, the harmonic balance method (HBM), a numerical method mainly working in the frequency domain, has been applied to solve a simple nonlinear clarinet model consisting of a linear exciter (for the reed) nonlinearly coupled to a linear resonator with visco-thermal losses (for the pipe). A recent and improved implementation of the HBM for self-sustained instruments has allowed us to study the model theoretically when including dispersion in the pipe or mass and damping terms in the reed model. The resulting periodic solutions for the internal pressure spectrum and the corresponding playing frequency are shown to align well with previous theoretical and experimental knowledge of the clarinet. Finally, we present and briefly discuss a few (probably unstable) oscillation regimes both with the HBM and with a real clarinet.     

具有几个光振荡周期的飞秒激光脉冲

从麦克斯韦方程组出发 ,推导出了具有几个光振荡周期的飞秒激光脉冲在非线性光纤中传输的方程和非线性光纤的折射率。给出了描述具有几个光振荡周期的飞秒激光脉冲在非线性光纤中传输方程的解。研究了在非线性光纤中自相位调制导致具有几个光振荡周期的飞秒激光脉冲频谱展宽 (脉宽压缩 )的详细物理过程。研究了非线性光纤中飞秒光孤子产生的条件     

Homotopic method of solving a class of EI Nino/La Nina-Southern Oscillation sea-air oscillator

The EI Nino/La Nina-Southern Oscillation （ENSO） is an interannual phenomenon involved in the tropical Pacific Ocean-atmosphere interactions. In this paper, an asymptotic method of solving the nonlinear equation for the ENSO model is created. And based on a class of oscillator of the ENSO model, the approximate solution of a corresponding problem is studied by employing the method of homotopic mapping. It is proved from the results that the homotopic method can be used for analysing the sea surface temperature anomaly in the equatorial eastern Pacific and the thermocline depth anomaly of the atmosphere-ocean oscillation for the ENSO model.     

High-power coaxial microwave ubitron: Simulation by the particle-in-cell method

Results of complete numerical simulation of a high-power coaxial microwave ubitron are reported. The simulation uses a 2.5-dimensional numerical electromagnetic code based on the particle-in-cell method and is performed for the amplification and oscillation operating modes. The results of our numerical simulation are compared with analytical results and results following from the nonlinear theory that are obtained by partial computer simulation.     

Perturbation method of studying the EI Nifio oscillation with two parameters by using the delay sea-air oscillator model

The EI Nio-southern oscillation (ENSO) is an interannual phenomenon involved in tropical Pacific oceanatmosphere interactions. In this paper, we develop an asymptotic method of solving the nonlinear equation using the ENSO model. Based on a class of the oscillator of the ENSO model, a approximate solution of the corresponding problem is studied employing the perturbation method.     

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.     

Perturbation method of studying the EI Niño oscillation with two parameters using delay sea—air oscillator model

The EI Niño-southern oscillation (ENSO) is an interannual phenomenon involved in the tropical Pacific ocean-atmosphere interactions. In this paper, we develop an asymptotic method of solving the nonlinear equation using the ENSO model. Based on a class of oscillator of the ENSO model, a approximate solution of the corresponding problem is studied employing the perturbation method.     

Effective medium method of slightly compressible elastic media permeated with air-filled bubbles

An effective medium method is developed for the slightly compressible elastic media permeated with air-filled bubbles, according to the nonlinear oscillation of the bubble, which happens when compressional wave travels through the porous media. The effective Lame coefficients of the porous medium and the nonlinear elastic wave equation are deduced, based on the fact that the micro-unit of the effective medium should have the same stress and strain as the micro-unit of the porous media. The linearized properties obtained by this method are in good agreement with the results of Gaunaurd’s classic theory [Gaunaurd G.C. and überall H., J. Acoust. Soc. Am., 1978, 63: 1699–1711]. Furthermore, the nonlinear coefficient, which is an important property of the porous media, can also be acquired by this method. __________ Translated from Acta Acustica, 2006 (in Chinese) (in press)     

Asymptotic solving method for a seaben air oscillator model of atmospheric physics

In this paper, a class of coupled system for the El Ni no/La Ni na southern oscillation (ENSO) atmospheric physics oscillation model is considered. We propose an ENSO atmospheric physics model using a method from the asymptotic theory. It is indicated from the results that the asymptotic method can be used for analyzing the sea surface temperature anomaly and the thermocline depth anomaly of the atmosphere-ocean oscillation for the ENSO model in the equatorial Pacific.     

Enhancement of Available Conversion Efficiency of Optical Parametric Amplifier in a Cascaded Photonic Crystal Structure

Using the cascaded structure of a linear and a second-order nonlinear photonic crystals, we realize a high-efficiency optical parametric amplifier in the case of exact phase matching. This proposal is verified using the slow-envelope nonlinear finite difference time domain numerical method. Compared with the case of the individual nonlinear photonic crystal structure, the oscillation threshold is decreased obviously; and the peak power amplification factor of the transmitted signal is enhanced more than 20 times.     

Projection-free proper orthogonal decomposition method for a cantilever plate in supersonic flow

This paper explores the use of the proper orthogonal decomposition (POD) method for supersonic nonlinear flutter of a cantilever plate or wing. The aeroelastic equations are constructed using von Karman plate theory and first-order piston theory. The two-dimensional POD modes (POMs) in xy plane are determined from the chaotic results given by the traditional Rayleigh–Ritz (RR) approach. For a specific structure, the POMs need to be calculated once and then can be used for various parameters of interest. The derivatives of the POMs are calculated numerically to avoid the complex projection from the POMs to the Rayleigh–Ritz modes (RRMs). Numerical examples demonstrate that the POD method using 4 POMs can obtain accurate limit cycle oscillation (LCO) results with substantial computational cost savings, compared with 12 RRMs by the Rayleigh–Ritz method. The POD method is employed for the analysis of the chaotic oscillations. It is also demonstrated that the POD modes are robust over a range of flight parameters.     

基于分数阶可停振动系统的周期未知微弱信号检测方法

本文建立了分数阶可停振动系统, 其可停振动状态的改变对周期策动力敏感, 对零均值随机微小扰动不敏感, 这事实上为周期未知微弱信号检测提供了一种新的高效检测方法和判别标准. 与现有的利用混沌系统的大尺度周期状态变化检测周期未知弱信号的方法 需逐一尝试设置不同频率内置信号以便期望与待检周期信号发生共振不同, 利用分数阶可停振动系统的可停振动状态变化检测周期未知微弱信号的方法, 除了同样具有因为状态变化对周期信号的敏感性而能够实现极低检测门限的特点外, 还具有混沌系统信号检测所不具有的优点: 1)无需预先估计待检信号的周期; 2)无需计算系统状态的临界阈值; 3)可停振动状态可由本文设计的指数波动函数可靠地进行判断; 4)通过系统微分阶数的变化, 将检测系统层次化, 从而可得到比整数阶检测系统更低的检测门限, 特别是在色噪声环境下, 通过选取合适的微分阶数, 基于分数阶可停振动系统的微弱周期信号检测法能够大幅度的降低检测门限, 在本文的仿真试验中, 检测门限可达-182 dB. 关键词： 分数阶非线性系统 Duffing振子 弱信号检测     

A RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in Lagrangian coordinate

In this paper, Runge-Kutta Discontinuous Galerkin (RKDG) finite element method is presented to solve the one-dimensional inviscid compressible gas dynamic equations in Lagrangian coordinate. The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method. A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method. For multi-medium fluid simulation, the two cells adjacent to the interface are treated differently from other cells. At first, a linear Riemann solver is applied to calculate the numerical flux at the interface. Numerical examples show that there is some oscillation in the vicinity of the interface. Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical flux at the interface, which suppress the oscillation successfully. Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.