The Dissipative Nonlocal Oscillator

The most important characteristics of a nonlocal and nonlinear oscillator subject to dissipative forces are extensively studied by means of an asymptotic perturbation method, based upon temporal rescaling and harmonic balance. The conditions under which bifurcations and limit cycles appear are determined. If the parameters satisfy particular conditions, a quasi-periodic motion is predicted, because a second small frequency adds to the natural frequency of the oscillator. The analytical results are validated by numerically solving the original system.     

Quasi-Periodic Oscillations,Chaos and Suppression of Chaos in a Nonlinear Oscillator Driven by Parametric and External Excitations

An analysis is given of the dynamic of a one-degree-of-freedom oscillator with quadratic and cubic nonlinearities subjected to parametric and external excitations having incommensurate frequencies. A new method is given for constructing an asymptotic expansion of the quasi-periodic solutions. The generalized averaging method is first applied to reduce the original quasi-periodically driven system to a periodically driven one. This method can be viewed as an adaptation to quasi-periodic systems of the technique developed by Bogolioubov and Mitropolsky for periodically driven ones. To approximate the periodic solutions of the reduced periodically driven system, corresponding to the quasi-periodic solution of the original one, multiple-scale perturbation is applied in a second step. These periodic solutions are obtained by determining the steady-state response of the resulting autonomous amplitude-phase differential system. To study the onset of the chaotic dynamic of the original system, the Melnikov method is applied to the reduced periodically driven one. We also investigate the possibility of achieving a suitable system for the control of chaos by introducing a third harmonic parametric component into the cubic term of the system.     

Gauss白噪声外激下Rayleigh振子的平稳响应与首次穿越

研究了Rayleigh振子在Gauss白噪声外激下的平稳响应和首次穿越。首先利用随机平均法给出了系统随机平均It^o微分方程,对平均方程的稳态概率密度做了数值分析;然后建立了条件可靠性函数的后向Kolmogorov方程及首次穿越时间条件矩的Pontragin方程;最后对三组不同的参数值分析了首次穿越的概率统计特性。     

谐和与随机噪声联合作用下非线性系统的响应

研究了Duffing振子在谐和与随噪声联合激励下的响应和稳应性问题。用谐波平均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晌应的影响。在一定条件下,系统具有两个均方响应值和跳跃现象。数值模拟表明本文提出的方法是有效的。     

Nonlinear Response of Cylindrical Shells to Random Excitation

An analytical study of nonlinear flexural vibrations of cylindrical shells to random excitation is presented. Donnell's thin-shell theory is used to develop the governing equations of motion. Thermal effects for a uniform temperature rise through the shell thickness are included in the formulation. A Monte Carlo simulation technique of stationary random processes, multi-mode Galerkin-like approach and numerical integration procedures are used to develop nonlinear response solutions of simply-supported cylindrical shells. Numerical results include time domain response histories, root-mean-square values and histograms of probability density. Comparison of Monte Carlo results is made to those obtained by statistical linearization and the Fokker–Planck equation.     

Normal Modes and Boundary Layers for a Slender Tensioned Beam on a Nonlinear Foundation

In this paper, the nonlinear normal modes (NNMs) of a thin beamresting on a nonlinear spring bed subjected to an axial tension isstudied. An energy-based method is used to obtain NNMs. In conjunction with amatched asymptotic expansion, we analyze, through simple formulas, thelocal effects that a small bending stiffness has on the dynamics, alongwith the secular effects caused by a symmetric nonlinearity. Nonlinearmode shapes are computed and compared with those of the unperturbedlinear system. A double asymptotic expansion is employed to compute theboundary layers in the nonlinear mode shape due to the small bendingstiffness. Satisfactory agreement between the theoretical and numericalbackbone curves of the system in the frequency domain is observed.     

Response Statistics of Three-Degree-of-Freedom Nonlinear System to Narrow-Band Random Excitation

The principal resonance of a 3-DOF nonlinear system to narrow-band random external excitations is investigated. The method of multiple scales is used to derive the equations for modulation of amplitude and phase. The behavior, stability and bifurcation of steady-state responses are studied by means of qualitative analysis. The effects of damping, detuning, and excitation intensity on responses are analyzed. The theoretical analyses are verified by numerical results. Both theoretical analyses and numerical simulations show that when the intensity of the random excitation increases, the nontrivial steady state solution may change from a limit cycle to a diffused limit cycle. Under some conditions, co-existence of two kinds of stable steady-state solutions, saturation and jump phenomena may occur. The stationary probability density function of responses for the co-existence case is obtained approximately.     

Experimental Nonlinear Identification of a Single Mode of a Transversely Excited Beam

A procedure is presented for using a primary resonance excitation in experimentally identifying the nonlinear parameters of a model approximating the response of a cantilevered beam by a single mode. The model accounts for cubic inertia and stiffness nonlinearities and quadratic damping. The method of multiple scales is used to determine the frequency-response function for the system. Experimental frequency- and amplitude-sweep data is compared with the prediction of the frequency-response function in a least-squares curve-fitting algorithm. The algorithm is improved by making use of experimentally known information about the location of the bifurcation points. The method is validated by using the extracted parameters to predict the force-response curves at other nearby frequencies.We then compare this technique with two other techniques that have been presented in the literature. In addition to the amplitude- and frequency-sweep technique presented, we apply a backbone curve- fitting technique and a time-domain technique to the second mode of a cantilevered beam. Differences in the parameter estimates are discussed. We conclude by discussing the limitations encountered for each technique. These include the inability to separate the nonlinear curvature and inertia effects and problems in estimating the coefficients of small terms with the time-domain technique.     

Effect of External Excitations on a Nonlinear System with Time Delay

The trivial equilibrium of a two-degree-of-freedom autonomous system may become unstable via a Hopf bifurcation of multiplicity two and give rise to oscillatory bifurcating solutions, due to presence of a time delay in the linear and nonlinear terms. The effect of external excitations on the dynamic behaviour of the corresponding non-autonomous system, after the Hopf bifurcation, is investigated based on the behaviour of solutions to the four-dimensional system of ordinary differential equations. The interaction between the Hopf bifurcating solutions and the high level excitations may induce a non-resonant or secondary resonance response, depending on the ratio of the frequency of bifurcating periodic motion to the frequency of external excitation. The first-order approximate periodic solutions for the non-resonant and super-harmonic resonance response are found to be in good agreement with those obtained by direct numerical integration of the delay differential equation. It is found that the non-resonant response may be either periodic or quasi-periodic. It is shown that the super-harmonic resonance response may exhibit periodic and quasi-periodic motions as well as a co-existence of two or three stable motions.     

窄带随机噪声激励下多自由度非线性系统的响应

在随机振动的研究中，研究较多的是系统在宽带噪声作用下的响应问题，对于非线性系统特别是多自由度非线性系统在窄带随机噪声作用下的响应问题则研究较少。本文研究了三自由度非线性系统在窄带随机噪声激励下的主共振响应和稳定性问题。用多尺度法分离了系统的快变项，给出了系统响应的振幅和相位角满足的方程。用摄动法讨论了系统随机项对系统响应的影响。当随机扰动较小时，在一定的参数范围内，对应于不同的初值，系统具有两个均方响应值，随机饱和现象也存在。当随机扰动增大时，系统可从一个大的响应突跳为一个小的响应，或从一个小的响应突跳为一个大的响应，即存在随机跳跃现象。数值模拟表明本文提出的方法是有效的。     

非线性结构动力学瞬态分析的摄动法

在用直接积分法求解非线性结构的动力响应时,常常需要做迭代运算。本文引入摄动方法后,加快了收敛速度,提高了计算效益。     

Nonlinear Vibrations of a Simply Supported Rectangular Metallic Plate Subjected to Transverse Harmonic Excitation in the Presence of a One-to-One Internal Resonance

The nonlinear response of rectangular and square metallic plates subjectto transverse harmonic excitations is studied. The nonlinearitiesoriginate from the use of Von Kármán strains. The method of multiplescales is used to solve the system of differential equationsapproximately. Frequency response curves are presented for both squareand rectangular plates for primary resonance of either mode in thepresence of a one-to-one internal resonance. Stability of steady statesolutions is investigated. Bifurcation points and their types arediscussed.     

The Asymptotic Perturbation Method for Nonlinear Continuous Systems

We apply the asymptotic perturbation (AP) method to the study of the vibrations of Euler--Bernoulli beam resting on a nonlinear elastic foundation. An external periodic excitation is in primary resonance or in subharmonic resonance in the order of one-half with an nth mode frequency. The AP method uses two different procedures for the solutions: introducing an asymptotic temporal rescaling and balancing the harmonic terms with a simple iteration. We obtain amplitude and phase modulation equations and determine external force-response and frequency-response curves. The validity of the method is highlighted by comparing the approximate solutions with the results of the numerical integration and multiple-scale methods.     

On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator

In this paper a strongly nonlinear forced oscillator will be studied. It will be shown that the recently developed perturbation method based on integrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how, in a rather efficient way, the existence and stability oftime-periodic solutions can be obtained from these approximations. In additionphase portraits, Poincaré-return maps, and bifurcation diagrams for a set of values of the parameters will be presented. In particularthe strongly nonlinear forced oscillator equation will be studied in this paper. It will be shown that the presentedperturbation method not onlycan be applied to a weakly nonlinear oscillator problem (that is, when the parameter ) but also to a strongly nonlinear problem (that is, when ). The model equation as considered in this paper is related to the phenomenon of galloping ofoverhead power transmission lines on which ice has accreted.     

Generalization of the method of full approximation and its applications

This paper presents a generalized form of the method of full approximation.By usingthe concept of asymptotic linearization and making the coordinate transformationsincluding the nonlinear functionals of dependent variables,the original nonlinear problemsare linearized and their higher-order solutions are given in terms of the first-termasymptotic solutions and corresponding transformations.The analysis of a model equationand some problems of weakly nonlinear oscillations and waves with the generalized methodshows that it is effective and straightforward.     

The Response of a Parametrically Excited van der Pol Oscillator to a Time Delay State Feedback

We investigate the parametric resonance of a van der Pol oscillator under state feedback control with a time delay. Using the asymptotic perturbation method, we obtain two slow-flow equations on the amplitude and phase ofthe oscillator. Their fixed points correspond to a periodic motion forthe starting system and we show parametric excitation-response andfrequency-response curves. We analyze the effect of time delay andfeedback gains from the viewpoint of vibration control and use energyconsiderations to study the existence and characteristics of limit cycles of the slow-flow equations. A limit cycle corresponds to a two-periodmodulated motion for the van der Pol oscillator. Analytical results areverified with numerical simulations. In order to exclude the possibilityof quasi-periodic motion and to reduce the amplitude peak of theparametric resonance, we find the appropriate choices for the feedbackgains and the time delay.     

具有小密度差的两层流体中运动点源的二阶内波解

在具有自由面的两层流体中，运动点源生成的Kelvin船波存在两种模式，即表面波模式和内波模式。当上、下层流体密度比趋于1时，由内波模式计算的界面波幅趋于无穷大，这与实验事实相违背。为克服此困难，在自由面和界面作小波幅运动的假设，引入一个小密度差参数。研究了运动点源在无粘、不可压且具有小密度差的两层有限深流体中生成的高阶波动。首先利用摄动方法推导了各阶小参数满足的边值问题；其次，给出了小密度差情形下的可解性条件。证明了在密度比趋于1的极限情形，不存在导致界面波幅无穷大的内波模式；最后，利用Phillips的非线性共振相互作用理论，构造了具有自由面的两层有限深流体中Kelvin船波系的二阶一致有效波动解，并证明了该解在深水情形下退化为Newman关于均匀流体中自由面的二阶波动解。     

Spectral Response of a Stochastic Oscillator under Impacts

An approximate analytical procedure is presented to estimate theresponse spectrum of an oscillator with elastic impacts under a Gaussian whitenoise excitation. The proposed approach is based on a perturbation analysis ofthe problem and on the use of the stochastic averaging principle. The basicidea is to replace the initial system by a more regular system obtained byapproximating the nonlinear restoring force by a Chebychev polynomial, and thento construct for this regular system two approximations: one for the flowand one for the stationary distribution of the response amplitude. Ananalytical approximation of the response spectrum can then be derived fromthese results. Predictions from this analytical approximation are compared with corresponding digital simulation estimates and with the ones obtained from theconventional equivalent linearization method.     

On the Nonlinear Dynamics of a Buckled Beam Subjected to a Primary-Resonance Excitation

We investigate the nonlinear response of a clamped-clamped buckled beamto a primary-resonance excitation of its first vibration mode. The beamis subjected to an axial force beyond the critical load of the firstbuckling mode and a transverse harmonic excitation. We solve thenonlinear buckling problem to determine the buckled configurations as afunction of the applied axial load. A Galerkin approximation is used todiscretize the nonlinear partial-differential equation governing themotion of the beam about its buckled configuration and obtain a set ofnonlinearly coupled ordinary-differential equations governing the timeevolution of the response. Single- and multi-mode Galerkinapproximations are used. We found out that using a single-modeapproximation leads to quantitative and qualitative errors in the staticand dynamic behaviors. To investigate the global dynamics, we use ashooting method to integrate the discretized equations and obtainperiodic orbits. The stability and bifurcations of the periodic orbitsare investigated using Floquet theory. The obtained theoretical resultsare in good qualitative agreement with the experimental results obtainedby Kreider and Nayfeh (Nonlinear Dynamics 15, 1998, 155–177.     

On Double Crater-Like Probability Density Functions of a Duffing Oscillator Subjected to Harmonic and Stochastic Excitation

It is a well-known phenomenon of the Duffing oscillator under harmonic excitation,that there is a frequency range, where two stable and one unstable stationarysolution coexist. If the Duffing oscillator is harmonically excited in thisfrequency range and additionally excited, e.g. by white noise, a double crater-likeprobability density function can be observed, if the noise intensity is smallcompared to the harmonic excitation. The aim of this paper is to calculate thisprobability density function approximately using perturbation techniques. Thestationary solutions in the deterministic case are calculated using theperturbation technique for the resonance case. In a second step, the probabilitydensity function of the perturbation of each of those stationary solutions iscalculated using the perturbation technique for the nonresonance case. This resultsin two crater-like probability density functions which are superimposed by usingthe probability of realization of each of the stationary solutions in thedeterministic case. The probability is calculated using numerical integration orthe method of slowly changing phase and amplitude. Finally, probability densityfunctions obtained in this manner are compared to Monte Carlo simulations.