On Mode Interactions for a Weakly Nonlinear Beam Equation

In this paper an initial-boundary value problem for a weakly nonlinear beam equation with a Rayleigh perturbation will be studied. It will be shown that the calculations to find internal resonances in this case are much more complicated than and differ substantially from the calculations for the weakly nonlinear wave equation with a Rayleigh perturbation as for instance presented in [3] or [7]. The initial-boundary value problem can be regarded as a simple model describing wind-induced oscillations of flexible structures like suspension bridges or iced overhead transmission lines. Using a two-timescales perturbation method approximations for solutions of this initial-boundary value problem will be constructed.     

On the Vibrations of a Simply Supported Square Plate on a Weakly Nonlinear Elastic Foundation

In this paper an initial-boundary value problem for a weakly nonlinear plate equation with a quadratic nonlinearity will be studied. This initial-boundary value problem can be regarded as a simple model describing free oscillations of a simply supported square plate on an elastic foundation. It is assumed that the foundation has a different behavior for compression and for expansion. An approximation for the solution of the initial-boundary value problem will be constructed using a two-timescales perturbation method. The existence and uniqueness of the solution of the problem will be proved. Also the asymptotic validity of the constructed approximations will be shown on long timescales. For specific parameter values, it turns out that complicated internal resonances occur.     

Forced oscillations of a vertical continuous rotor with geometric nonlinearity

Nonlinear forced oscillations of a vertical continuous rotor with distributed mass are discussed. The restoring force of the rotor has geometric stiffening nonlinearity due to the extension of the rotor center line. The possibility of the occurrence of nonlinear forced oscillations at various subcritical speeds and the shapes of resonance curves at the major critical speeds and at some subcritical speeds are investigated theoretically. Consequently, the following is clarified: (a) the shape of resonance curves at the major critical speed becomes a hard spring type, and (b) among various kinds of nonlinear forced oscillations, only some special kinds of combination resonances have possibility of occurrence.     

Nonlinear time transformation method for strong nonlinear oscillation systems

In this paper, a nonlinear time transformation method is presented for the analysis of strong nonlinear oscillation systems. This method can be used to study the limit cycle behavior of the autonomous systems and to analyze the forced vibration of a strong nonlinear system. The project partly supported by the Foundation of Zhongshan University Advanced Research Center     

1/3 subharmonic solution of elliptical sandwich plates

IntroductionInrecentyears,withtheessentialadvantageoflightweightandhighrigidity ,sandwichplatesandshellshavebeenusedasanimportantpatternofstructuralelementsinaeronautical,astronauticalandnavalengineering .However,nonlinearproblemsforsandwichplatesandshellsareonlyinvestigatedbyafewbecauseofthedifficultiesofnonlinearmathematicalproblems.LiuRen_huaiandXuJia_chu[1,2 ]andothershavemadesomeinvestigationsinthisfield .Bifurcationofnonlinearvibrationforsandwichplateshasnotyetbeeninvestigated .Inthisp…     

Forced Oscillations of a System,Containing a Snap-Through Truss,Close to Its Equilibrium Position

The nonlinear dynamics of a two-degree-of-freedom mechanical system is considered. This system consists of a linear oscillator under the action of a time-periodic force and a snap-through truss, which acts as an absorber of the forced oscillations of the linear main system. The forced oscillations of the snap-through truss close to its equilibrium position are analyzed by the multiple scales method.     

Enhanced Nonlinear Dynamics for Accurate Identification of Stiffness Loss in a Thermo-Shielding Panel

The dynamics of a panel forced by transverse loads and undergoing limit cycle oscillations and chaos is investigated. The nonlinear von Karman plate theory is used to obtain a model for healthy and damaged panels. Damage is modeled by a loss of stiffness in a portion of the plate. The presence of low levels of damage is identified by using an external nonlinear excitation and analyzing the attractor of the resulting dynamics in state space. Most of the current studies of such problems are based on linear theories and linear structures. In contrast, the results presented are obtained by using and enhancing nonlinear and chaotic dynamics, and have the advantage of an increased accuracy in detecting damage and monitoring structural health.An earlier version of this paper was presented at the 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Palm Springs, California, April 2004.     

Controlling chaotic oscillations of visco-elastic plates by the linearization via output feedback

Ｃｏｎｔｒｏｌｌｉｎｇｃｈａｏｓｈａｓｄｒａｗｎｉｎｃｒｅａｓｉｎｇａｔｔｅｎｔｉｏｎｂｅｃａｕｓｅｏｆｉｔｓｔｈｅｏｒｅｔｉｃａｌｉｍｐｏｒｔａｎｃｅａｎｄｐｏｓｓｉｂｌｅａｐｐｌｉｃａｔｉｏｎｓ,ａｎｄｍｕｃｈｐｒｏｇｒｅｓｓｈａｓｂｅｅｎａｃｈｉｅｖｅｄ［１～４］．Ｔｈｅｅｘａｃｔｌｉｎｅａｒｉｚａｔｉｏｎｉｓａｎｉｍｐｏｒｔａｎｔａｐｐｒｏａｃｈｔｏａｎａｌｙｚｅａｎｄｄｅｓｉｇｎｎｏｎｌｉｎｅａｒｃｏｎｔｒｏｌｓｙｓｔｅｍｓ,ａｎｄｈａｓｂｅｅｎｅｍｐｌｏｙｅｄｔｏｃｏｎｔｒｏｌｃｈａ…     

Ellipsoidal Oscillations of a Gas Bubble with Periodic Variation of the Surrounding Fluid Pressure

Ellipsoidal linear and nonlinear oscillations of a gas bubble under harmonic variation of the surrounding fluid pressure are studied. The system is considered under conditions in which periodic sonoluminescence of the individual bubble in a standing acoustic wave is observable. A mathematical model of the bubble dynamics is suggested; in this model, the variation of the gas/fluid interface shape is described correct to the square of the amplitude of the deformation of the spherical shape of the bubble. The character of the air bubble oscillations in water is investigated in relation to the initial bubble radius and the fluid pressure variation amplitude. It is shown that nonspherical oscillations of limited amplitude can occur outside the range of linearly stable spherical oscillations. In this case, both oscillations with a period equal to one or two periods of the fluid pressure variation and aperiodic oscillations can be observed.     

Resonance,Parameter Estimation,and Modal Interactions in a Strongly Nonlinear Benchtop Oscillator

We study the vibrations of a strongly nonlinear, electromechanically forced, benchtop experimental oscillator. We consciously avoid first-principles derivations of the governing equations, with an eye towards more complex practical applications where such derivations are difficult. Instead, we spend our effort in using simple insights from the subject of nonlinear oscillations to develop a quantitatively accurate model for the single-mode resonant behavior of our oscillator. In particular, we assume an SDOF model for the oscillator; and develop a structure for, and estimate the parameters of, this model. We validate the model thus obtained against experimental free and forced vibration data. We find that, although the qualitative dynamics is simple, some effort in the modeling is needed to quantitatively capture the dynamic response well. We also briefly study the higher dimensional dynamics of the oscillator, and present some experimental results showing modal interactions through a 0:1 internal resonance, which has been studied elsewhere. The novelty here lies in the strong nonlinearity of the slow mode.     

Nonsmall liquid oscillations in moving vessels

The system of approximate nonlinear equations describing liquid oscillations in axisymmetric vessels is constructed. The equations are obtained for the case in which two coordinates belonging to the family of generalized coordinates characterizing the liquid motion are not small. This family is selected so that from the resulting nonlinear equations we can obtain as a particular case the nonlinear equations of [1–3], which are valid for the class of cylindrical vessels, and the requirements are satisfied that the resulting nonlinear equations correspond to the widely adopted linearized equations of liquid oscillations [4–6], Nonlinear equations are obtained which describe liquid oscillations in arbitrary vessels of rotation with radial baffles.     

Absorbing Balls for Equations Modeling Nonuniform Deformable Bodies with Heavy Rigid Attachments

We study degenerate nonlinear partial differential equations with dynamical boundary conditions describing the forced motions of nonuniform deformable bodies with heavy rigid attachments. We prove that the dynamical system generated by a discretization of these equations has an absorbing ball whose size is independent of the order of the discretization. This result implies the existence of an absorbing ball for the infinite-dimensional dynamical system corresponding to the original degenerate partial differential equation and thereby serves as a critical step for establishing the existence of global attractors for this system. Our results also address the interesting mechanical question of how nonuniformity complicates the longterm dynamics of the coupled systems we consider.     

Analytic evaluation of periodic responses of a forced nonlinear oscillator

The periodic responses of a strongly nonlinear, single-degree-of-freedom forced oscillator with weak excitation and damping are examined. The presented methodology is based on a regular perturbation expansion, whose first term is the solution of the unforced, and undamped nonlinear problem. Higher order approximations are computed by explicitly solving linear differential equations possessing a periodically varying coefficient. The general theory is used for studying the periodic steady state motions of the periodically forced system. Moreover, it is shown that the presented analysis can be used to analytically study the orbital stability of the identified steady state motions. The proposed method can also be used for studying periodic responses due to nonperiodic transient forces, provided that these responses are close to the O(1) periodic generating solution.     

Influence of heat transfer on the dynamic response of a spherical gas/vapour bubble

The standard approach to analyse the bubble motion is the well known Rayleigh–Plesset equation. When applying the toolbox of nonlinear dynamical systems to this problem several aspects of physical modelling are usually sacrificed. Particularly in vapour bubbles the heat transfer in the liquid domain has a significant effect on the bubble motion; therefore the nonlinear energy equation coupled with the Rayleigh–Plesset equation must be solved. The main aim of this paper is to find an efficient numerical method to transform the energy equation into an ODE system, which, after coupling with the Rayleigh–Plesset equation can be analysed with the help of bifurcation theory. Due to the strong nonlinearity and violent bubble motions the computational effort can be high, thus it is essential to reduce the size of the problem as much as possible. In the first part of the paper finite difference, Galerkin and spectral collocation methods are examined and compared in terms of efficiency. In the second part free and forced oscillations are analysed with an emphasis on the influence of heat transfer. In the case of forced oscillations the unstable branches of the amplification diagrams are also computed.     

A Modified Exact Linearization Control for Chaotic Oscillators

The control of chaotic oscillations is investigated in this paper. A control methodology, termed input-output linearization, is modified by locally linearizing the nonlinear control law in the small neighborhood of the control goal. Its suitability for controlling chaotic oscillators is analyzed. The forced Duffing oscillator is treated as a numerical example of controlling chaotic motion to a given fixed point and a given period-2 motion. The control signals and time needed to achieve the desired goals of the modified method are compared with those of the original method. The robustness of the control law is demonstrated.     

Nonlinear waves in fluid flow through a viscoelastic tube

A system of nonlinear equations for describing the perturbations of the pressure and radius in fluid flow through a viscoelastic tube is derived. A differential relation between the pressure and the radius of a viscoelastic tube through which fluid flows is obtained. Nonlinear evolutionary equations for describing perturbations of the pressure and radius in fluid flow are derived. It is shown that the Burgers equation, the Korteweg-de Vries equation, and the nonlinear fourth-order evolutionary equation can be used for describing the pressure pulses on various scales. Exact solutions of the equations obtained are discussed. The numerical solutions described by the Burgers equation and the nonlinear fourth-order evolutionary equation are compared.     

Articulated Pipes Conveying Fluid Pulsating with High Frequency

Stability and nonlinear dynamics of two articulated pipes conveying fluid with a high-frequency pulsating component is investigated. The non-autonomous model equations are converted into autonomous equations by approximating the fast excitation terms with slowly varying terms. The downward hanging pipe position will lose stability if the mean flow speed exceeds a certain critical value. Adding a pulsating component to the fluid flow is shown to stabilize the hanging position for high values of the ratio between fluid and pipe-mass, and to marginally destabilize this position for low ratios. An approximate nonlinear solution for small-amplitude flutter oscillations is obtained using a fifth-order multiple scales perturbation method, and large-amplitude oscillations are examined by numerical integration of the autonomous model equations, using a path-following algorithm. The pulsating fluid component is shown to affect the nonlinear behavior of the system, e.g. bifurcation types can change from supercritical to subcritical, creating several coexisting stable solutions and also anti-symmetrical flutter may appear.     

求解非线性动力系统周期解推广的打靶法

提出一种确定非线性系统周期轨道及周期的改进打靶算法。首先通过改变系统的时间尺度，将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中，然后对传统打靶法进行改造，将周期也作为一个参数一起参入打靶法的迭代过程，从而能迅速确定出系统的周期轨道及其周期。该方法对初始迭代参数没有苛刻要求，可以用于分析强非线性系统，而且对参数激励系统同样有效，对高维系统也能迅速、准确地求得周期解。文中应用该方法对三维Rǒssler系统和八维非线性柔性转子-轴承系统的周期轨道和周期进行了求解，通过与四阶Runge-Kutta数值积分结果比较，验证了方法的有效性。     

Longwave Approximation in Film Flow Theory

An asymptotic longwave model which takes dispersive terms into account is constructed for describing the motion of thin films with finite deviations from the middle surface. An exact periodic solution describing a nonlinear capillary wave is constructed within the framework of the model. Small deviations from the nonlinear capillary wave are described by a linear system with periodic coefficients. It is shown that for wave perturbation periods greater than a certain critical value the monodromy matrix of this system has eigenvalues whose absolute values are equal to unity. For perturbation periods less than the critical period the absolute value of one of the eigenvalues becomes greater than unity.     

An investigation on primary resonance phenomena of elastic medium based single walled carbon nanotubes

In this paper, the geometrically nonlinear free and forced oscillations of simply supported single walled carbon nanotubes (SWCNTs) are analytically investigated on the basis of the Euler–Bernoulli beam theory. The nonlinear frequencies of SWCNTs with initial lateral displacement are discussed. Equations have been solved using an exact method for free vibration and multiple times scales (MTS) method for forced vibration and some analytical relations have been obtained for natural frequency of oscillations. The numerical results reveal that the nonlinear free and forced vibration of nanotubes is effected significantly by both surrounding elastic medium and CNT aspect ratio.