A Method for Obtaining Approximate Analytic Periods for a Class of Nonlinear Oscillators

This paper deals with nonlinear oscillations of conservative single-degree-of-freedom systems with odd nonlinearity. By combining the linearization of the governing equation with the method of harmonic balance, we establish two approximate analytic formulas for the period. These two formulas are valid for small as well as large amplitudes of oscillation. Three examples are used to illustrate that the proposed formulas can give very satisfactory approximate results. Sommario. Questo lavoro tratta il problema delle oscillazioni nonlineari di sistemi conservativi ad un grado di libertà con nonlinearità simmetriche. Combinando opportunamente la tecnica di linearizzazione dellequazione del moto con il metodo del bilancio armonico si perviene a due formule analitiche approssimate per il periodo. Le formule ottenute sono valide sia per piccole che per grandi ampiezze di oscillazione. Si utilizzano tre esempi classici di oscillatori nonlineari per illustrate lefficacia del metodo nel produrre risultati approssimati soddisfacenti.     

ASYMPTOTIC ANALYSIS OF A CLASS OF NONLINEAR OSCILLATION EQUATION IN ELECTRICAL ENGINEERING

ＡＳＹＭＰＴＯＴＩＣＡＮＡＬＹＳＩＳＯＦＡＣＬＡＳＳＯＦＮＯＮＬＩＮＥＡＲＯＳＣＩＬＬＡＴＩＯＮＥＱＵＡＴＩＯＮＩＮＥＬＥＣＴＲＩＣＡＬＥＮＧＩＮＥＥＲＩＮＧＣｈｅｎｇＹｏｕ－ｌｉａｎｇ（程友良）（ＤｅｐａｒｔｍｅｎｔｏｆＦｕｎｄａｍｅｎｔａｌＣｏｕ...     

Limit cycle analysis of a class of strongly nonlinear oscillation equations

The limit cycle of a class of strongly nonlinear oscillation equations of the form % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqadwhagaWaaiabgUcaRmXvP5wqonvsaeHbbjxAHXgiofMCY92D% aGqbciab-DgaNjab-HcaOiaadwhacqWFPaqkcqWF9aqpcqaH1oqzca% WGMbGaaiikaiaadwhacaGGSaGabmyDayaacaGaaiykaaaa!50B8!\[\ddot u + g(u) = \varepsilon f(u,\dot u)\] is investigated by means of a modified version of the KBM method, where is a positive small parameter. The advantage of our method is its straightforwardness and effectiveness, which is suitable for the above equation, where g(u) need not be restricted to an odd function of u, provided that the reduced equation, corresponding to =0, has a periodic solution. A specific example is presented to demonstrate the validity and accuracy of our 09 method by comparing our results with numerical ones, which are in good agreement with each other even for relatively large .     

NONLINEAR ANALYSIS OF EQUATORIAL EASTERN PACIFIC AIR-SEA COUPLING OSCILLATION AND A LIMIT- CYCLE THEORY FOR ENSO CYCLE

The troposphere and ocean mixed layer were considered as two components of a dynamic system operated by solar radiation as the constant source of energy, where upon an air-sea coupling self-exited coupling oscillation model was based with the aid of a locally averaged thermodynamic climate model, resulting mathematically in a closed self-governed dynamic system, a so-called El Nino-Southern Oscillation(ENSO) system. With the limit cycle solution of the system. It is shown that the essential physics of the coupled system can be described by the ENSO system. Compared with the observations, the theoretical limit cycle orbit matches the observed phase loop qualitatively. The ENSO system provides a useful theoretical framework for study of interannual variation of the tropical climate system.     

Energy Method for Approximate Periodic Solution of Strongly Nonlinear Nonautonomous Systems

In this paper, based upon the concept of the conservation of average energy, a theorem for the periodic solution of strongly nonlinear nonautonomous systems is proved. By using this theorem not only the necessary and sufficient conditions for the existence and stability of the periodic solutions can be obtained but, at the same time, the approximate expressions for those periodic solutions can also be obtained.     

Output Feedback Modular Adaptive Control of a Nonlinear Prototypical Wing Section

The paper presents nonlinear adaptive control systems for the control of limit cycle oscillations of a prototypical wing section with structural nonlinearities using only output feedback. The chosen model describes the plunge and pitch motion of a wing. The model includes plunge and pitch nonlinearities, and has a single control surface for the purpose of control. Using a canonical representation of the aeroelastic system, a modular output feedback adaptive control system consisting of an input-to-state stabilizing controller and a passive identifier (an observer and adaptation law) is derived. In the closed-loop system, asymptotic stabilization of the pitch and plunge motion is accomplished. Simulation results show that the control system is effective in regulating the state vector to the origin in spite of large parameter uncertainties.     

Energy Based Stochastic Estimation of Nonlinear Oscillators with Parametric Random Excitation

A new energy-based system identification method is developed, applicable in situations where the dynamic response of a structure is measurable but the excitation is unmeasurable and describable only in terms of a stochastic process. It is shown that, in the case of a non-linear single degree of freedom system subjected to purely parametric, non-white random excitation, the power spectrum of the excitation can be identified through an estimation of the diffusion coefficient relating to the energy envelope of the response process. Through an estimation of the drift coefficient an identification of the system damping is also possible. The method is validated through application to simulated data relating to a Duffing oscillator with non-linear damping.     

Free Vibration in a Class of Self-Excited Oscillators with 1:3 Internal Resonance

Free vibration of a two degree of freedom weakly nonlinear oscillator is investigated. The type of nonlinearity considered is symmetric, it involves displacement as well as velocity terms and gives rise to self-excited oscillations in many engineering applications. After presenting the equations of motion in a general form, a perturbation methodology is applied for the case of 1:3 internal resonance. This yields a set of four slow-flow nonlinear equations, governing the amplitudes and phases of approximate motions of the system. It is then shown that these equations possess three distinct types of solutions, corresponding to trivial, single-mode and mixed-mode response of the system. The stability analysis of all these solutions is also performed. Next, numerical results are presented by applying this analysis to a specific practical example. Response diagrams are obtained for various combinations of the system parameters, in an effort to provide a complete picture of the dynamics and understand the transition from conditions of 1:3 internal resonance to non-resonant response. Emphasis is placed on identifying the effect of the linear damping, the frequency detuning and the stiffness nonlinearity parameters. Finally, the predictions of the approximate analysis are confirmed and extended further by direct integration of the averaged equations. This reveals the existence of other regular and irregular motions and illustrates the transition from phase-locked to drift response, which takes place through a Hopf bifurcation and a homoclinic explosion of the averaged equations.     

Nonlinear oscillations with parametric excitation solved by homotopy analysis method

An analytical technique, namely the homotopy analysis method （HAM）, is used to solve problems of nonlinear oscillations with parametric excitation. Unlike perturbation methods, HAM is not dependent on any small physical parameters at all, and thus valid for both weakly and strongly nonlinear problems. In addition, HAM is different from all other analytic techniques in providing a simple way to adjust and control convergence region of the series solution by means of an auxiliary parameter h. In the present paper, a periodic analytic approximations for nonlinear oscillations with parametric excitation are obtained by using HAM, and the results are validated by numerical simulations.     

一种确定非线性振子多个共存周期解的数值方法

提出一种简洁高效数值方法，用以确定周期激励下非线性振子的多个共存周期响应。这是一种基于Ｐｏｉｎｃａｒé映射和二维自治系统奇点分类的几何方法，可在Ｐｏｉｎｃａｒé截面上得出多个周期运动的分布情况和类型。文中二个算例证实了该方法的有效性。     

Approximate First Integrals of Weakly Nonlinear,Damped-Driven Oscillators with One Degree of Freedom

First-order approximate symmetries of weaklynonlinear, damped-driven oscillators have been determined.First-order approximate first integrals have been obtained byemploying an approximate version of Noether's theorem for the conservativecase. Furthermore, approximate first integrals of the damped case have been obtained based on the first integrals of the conservative case.Approximate first integrals enabled us to identify three types of genericbifurcations. Analytical results have been verified by numerical experiments.     

On oscillations of a frictional pendulum

Conditions are established under which a standard limit cycle occurs in the system under consideration, or the trajectory closes under the influence of a stagnation domain. It is pointed out that when the solution falls into the stagnation domain it makes no sense to use the asymptotic method because of a large error __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 104–112, February 2006.     

Subharmonic Response of a Quasi-Isochronous Vibroimpact System to a Randomly Disordered Periodic Excitation

A quasi-isochronous vibroimpact system is considered, i.e. a linear system with a rigid one-sided barrier, which is slightly offset from the system's static equilibrium position. The system is excited by a sinusoidal force with disorder, or random phase modulation. The mean excitation frequency corresponds to a simple or subharmonic resonance, i.e. the value of its ratio to the natural frequency of the system without a barrier is close to some even integer. Influence of white-noise fluctuations of the instantaneous excitation frequency around its mean on the response is studied in this paper. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, or velocity jumps, thereby permitting the application of asymptotic averaging over the period for slowly varying inphase and quadrature responses. The averaged stochastic equations are solved exactly by the method of moments for the mean square response amplitude for the case of zero offset. A perturbation-based moment closure scheme is proposed for the case of nonzero offset and small random variations of amplitude. Therefore, the analytical results may be expected to be adequate for small values of excitation/system bandwidth ratio or for small intensities of the excitation frequency variations. However, at very large values of the parameter the results are approaching those predicted by a stochastic averaging method. Moreover, Monte-Carlo simulation has shown the moment closure results to be sufficiently accurate in general for any arbitrary bandwidth ratio. The basic conclusion, both of analytical and numerical simulation studies, is a sort of smearing of the amplitude frequency response curves owing to disorder, or random phase modulation: peak amplitudes may be strongly reduced, whereas somewhat increased response may be expected at large detunings, where response amplitudes to perfectly periodic excitation are relatively small.     

一类非线性系统在随机激励下的分叉

本文研究一类阻尼为线性，弹性恢复力为非线性的振动系统在随机外部激励作用下的随机分叉。文中采用广义稳态势和方法，求解系统响应的稳态联合概率密度函数。在此基础上根据由不变测度定义的随机分叉，讨论了具有权式分叉的确定性非线性系统在随机扰动下分叉行为。     

Attractor for a Degenerate Nonlinear Diffusion Problem with Nonlinear Boundary Condition

In this paper we prove the existence of a compact attractor in L () for a degenerate nonlinear diffusion problem with nonlinear flux on the boundary. In order to formulate the equation as a dynamical system, some existence and uniqueness results for weak solutions are proved.     

A Perturbation-Incremental Method for Strongly Nonlinear Autonomous Oscillators with Many Degrees of Freedom

The perturbation-incremental method is extended to determine thebifurcations and limit cycles of strongly nonlinear autonomousoscillators with many degrees of freedom. Coupled van der Poloscillators and coupled Rayleigh oscillators are taken as numericalexamples. Limit cycles of the oscillators can be calculated to anydesired degree of accuracy. The stabilities of limit cycles are alsodiscussed.     

时滞对于参数激励系统周期运动的影响

本文以结构物承受周期激励地震波和具有时滞的弹性地基为背景，研究时滞对于结构物振动响应的影响规律。问胚的数学模型是一个非线性参数激励时滞系统，采用中心流形和平均法得到Hopf分岔方程，研究结果表明时滞量对结构物的地震响应有重要影响，它可使响应振幅增大，意味着结构物遭破坏的危险性增大，同时，结果也表明可以通过阻尼控制器克服时滞引起的危险性。最后，对于不同的时滞量，比较理论分析与数值模拟结果基本吻合，说明了本文结果的可靠性。     

Bifurcation and stability of spatially periodic solutions of nonlinear evolution equations with integral operators

A more general kind of nonlinear evolution equations with integral operators is discussed in order to study the spatially periodic static bifurcating solutions and their stability. At first, the necessary condition and the sufficient condition for the existence of bifurcation are studied respectively. The stability of the equilibrium solutions is analyzed by the method of semigroups of linear operators. We also obtain the principle of exchange of stability in this case. As an example of application, a concrete result for a special case with integral operators of exponential type is presented. This work was supported by the Chinese National Foundation of Natural Science     

Asymptotic solution of Brusselator limit cycle in biochemistry

In this paper,two kinds of asymptotic analytic expression of the Brusselator limitcycle are given by means of two simple methods.To some extem,they are better thanthe analytic expression in [2].     

On the Determination of Nonlinear Response Distributions for Oscillators with Combined Harmonic and Random Excitation

The analysis of two different nonlinear systems, both subject to an excitation that comprises a harmonic and a random component, is presented in this paper. Both systems are known to exhibit different coexisting attractors for a purely harmonic forcing. The random part causes a disturbance of the response of these systems: Even though a dominating effect of the attractors for the deterministic case is still visible, the random disturbance also leads to occasional jumps between the areas surrounding the different attractors. To access the likelihood of the system being found in a specific state, probability density functions are approximated numerically by means of a localized statistical linearization.