Three-dimensional oscillations of suspended cables involving simultaneous internal resonances

The near resonant response of suspended, elastic cables driven by planar excitation is investigated using a three degree-of-freedom model. The model captures the interaction of a symmetric in-plane mode with two out-of-plane modes. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. For particular magnitudes of equilibrium curvature, the natural frequency of the in-plane mode is simultaneously commensurable with the natural frequencies of the two out-of-plane modes in 1:1 and 2:1 ratios. A second nonlinear order perturbation analysis is used to determine the existence and stability of four classes of periodic solutions. The perturbation solutions are compared with results obtained by numerically integrating the equations of motion. Furthermore, numerical simulations demonstrate the existence of quasiperiodic responses.A portion of this work was presented at the 1992 ASME Winter Annual Meeting, Anaheim, CA.     

Nonlinear oscillations of suspended cables containing a two-to-one internal resonance

The near-resonant response of suspended, elastic cables driven by planar excitation is investigated using a two degree-of-fredom model. The model captures the interaction of a symmetric in-plane mode and an out-of-plane mode with near commensurable natural frequencies in a 2:1 ratio. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. The existence and stability of periodic solutions are investigated using a second order perturbation analysis. The first order analysis shows that suspended cables may exhibit saturation and jump phenomena. The second order analysis, however, reveals that the cubic nonlinearities and higher order corrections disrupt saturation. The stable, steady state solutions for the second order analysis compare favorably with results obtained by numerically integrating the equations of motion.     

Forced response of quadratic nonlinear oscillator: comparison of various approaches

The primary resonances of a quadratic nonlinear system under weak and strong external excitations are investigated with the emphasis on the comparison of different analytical approximate approaches. The forced vibration of snap-through mechanism is treated as a quadratic nonlinear oscillator. The Lindstedt-Poincaré method, the multiple-scale method, the averaging method, and the harmonic balance method are used to determine the amplitude-frequency response relationships of the steady-state responses. It is demonstrated that the zeroth-order harmonic components should be accounted in the application of the harmonic balance method. The analytical approximations are compared with the numerical integrations in terms of the frequency response curves and the phase portraits. Supported by the numerical results, the harmonic balance method predicts that the quadratic nonlinearity bends the frequency response curves to the left. If the excitation amplitude is a second-order small quantity of the bookkeeping parameter, the steady-state responses predicted by the second-order approximation of the LindstedtPoincaré method and the multiple-scale method agree qualitatively with the numerical results. It is demonstrated that the quadratic nonlinear system implies softening type nonlinearity for any quadratic nonlinear coefficients.     

Random excitation of nonlinear coupled oscillators

This paper presents the experimental results of random excitation of a nonlinear two-degree-of-freedom system in the neighborhood of internal resonance. The random signals of the excitation and response coordinates are processed to estimate the mean squares, autocorrelation functions, power spectral densities, and probability density functions. The results are qualitatively compared with those predicted by the Fokker-Planck equation together with a non-Gaussian closure scheme. The effects of system damping ratios, nonlinear coupling parameter, internal detuning ratio, and excitation spectral density level are considered in both studies except the effect of damping ratios is not considered in the experimental investigation. Both studies reveal similar dynamic features such as autoparametric absorber effect and stochastic instability of the coupled system. The experimental results show that the autocorrelation function of the coupled system has the feature of ultra narrow band process and degenerates to a periodic one as the internal detuning departs from the exact internal resonance condition. The measured probability density functions of the response of the main system suggests that the Gaussian representation is sufticient as long as the excitation level is relatively low in the neighborhood of the system internal resonance condition.     

Nonlinear vibrations in beams and frames: The effect of the deformed equilibrium state

In the study of nonlinear vibrations of planar frames and beams with infinitesimal displacements and strains, the influence of the static displacements resulting from gravity effect and other conservative loads is usually disregarded. This paper discusses the effect of the deformed equilibrium configuration on the nonlinear vibrations through the analysis of two planar structures. Both structures present a two-to-one internal resonance and a primary response of the second mode. The equations of motion are reduced to two degrees of freedom and contain all geometrical and inertial nonlinear terms. These equations are derived by modal superposition with additional subsidiary conditions. In the two cases analyzed, the deformed equilibrium configuration virtually coincides with the undeformed configuration. Also, 2% is the maximum difference presented by the first two lower frequencies. The modes are practically coincident for the deformed and undeformed configurations. Nevertheless, the analysis of the frequency response curves clearly shows that the effect of the deformed equilibrium configuration produces a significant translation along the detuning factor axis. Such effect is even more important in the amplitude response curves. The phenomena represented by these curves may be distinct for the same excitation amplitude.     

Forced nonlinear oscillations of semi-infinite cables and beams resting on a unilateral elastic substrate

In this work, we study the nonlinear oscillations of mechanical systems resting on a (unilateral) elastic substrate reacting in compression only. We consider both semi-infinite cables and semi-infinite beams, subject to a constant distributed load and to a harmonic displacement applied to the finite boundary. Due to the nonlinearity of the substrate, the problem falls in the realm of free-boundary problems, because the position of the points where the system detaches from the substrate, called Touch Down Points (TDP), is not known in advance. By an appropriate change of variables, the problem is transformed into a fixed-boundary problem, which is successively approached by a perturbative expansion method. In order to detect the main mechanical phenomenon, terms up to the second order have to be considered. Two different regimes have been identified in the behaviour of the system, one below (called subcritical) and one above (called supercritical) a certain critical excitation frequency. In the latter, energy is lost by radiation at infinity, while in the former this phenomenon does not occur and various resonances are observed instead; their number depends on the statical configuration around which the system performs nonlinear oscillations.     

On nonlinear oscillations of a thin bar

Summary Based on one of the simplest mathematical model of a solid, nonlinear interactions between waves in a rectilinear bar are investigated, in order to reveal and display a number of dynamic properties inherent not only to the bar, but also to most weakly nonlinear mechanical systems with internal resonances. The presence of internal resonances in the bar is twofold. Firstly, there exists a slow periodic energy exchange between the longitudinal and the two quasi-harmonic bending waves involved in the resonant triad due to the phase matching, secondly, triple-frequency envelope solitons can be created from the resonant triad with the same modal state. The paper investigates the evolution of waves in the bar with the aim to classify the elementary type of wave triplet resonant interactions and define their existence and coesistence areas.The research described here has been made possible in part by Grant N R9B000 from the International Science Foundation. The authors would like to thank Professor G.A. Maugin for having sent copies of his papers, in particular [23], as well as for his permanent interest in our work.     

Parametric vibrations of flexible hoses excited by a pulsating fluid flow,Part I: Modelling,solution method and simulation

The article presents an analysis of a model describing lateral vibrations of a pipe induced by fluid flow velocity pulsation. The motion has been described with a set of two non-linear partial differential equations with periodically variable coefficients. In the analysis Galerkin method has been applied using orthogonal polynomials as shape function. To determine instability regions Floquet theory has been employed. The effect of selected parameters on parametric resonance ranges and regions of increased vibration level has been investigated. The character and form of vibrations have been investigated indicating the possibility of excitation of sub-harmonic and quasi-periodic vibrations in the combination resonance ranges.     

Effect of static loading on the nonlinear vibrations of a three-time redundant portal frame: Analytical and numerical studies

An analytical study of the nonlinear vibrations in a three-time redundant portal frame is presented herewith, considering the effect of the axial forces caused by the static loading upon the first anti-symmetrical mode (sway) and the first symmetrical mode natural frequencies. It is seen that the axial forces may play an important role in tuning the sway mode and the first symmetrical mode into a 1:2 internal resonance. Harmonic support excitations resonant with the first symmetrical mode are then introduced and the amplitudes of nonlinear steady states are computed based upon a multiple scales solution. Comparisons with numerical analyses using a finite-element program developed by the authors show good qualitative agreement.     

Free nonlinear oscillations of a thermally relaxing spherical gas bubble in an incompressible liquid

The quasi-adiabatic regime of free oscillation of a bubble in the presence of irreversible interphase heat transfer between the bubble and the ambient liquid is studied. On the basis of simplified model equations of a rarefield bubble mixture, a nonlinear-oscillation equation of the relaxation type is obtained. In constructing an exact particular solution of this equation, the heat transfer law associated with bubble compression is established. For studying the harmonic oscillations, the Krylov-Bogolyubov-Mitropol’skii asymptotic method is used. It is shown that, for a small bubble, the viscosity and heat transfer effects are of the same order. For a small bubble, the influence of these effects on the formation of the natural-oscillation frequency, which is small in the linear approximation, may be significant in the nonlinear formulation. For a large bubble, the influence of these effects is negligible in both approximations. For the approximate solution of the nonlinear equation, a uniformly valid second-order expansion is constructed.     

Analysis of a quadratic nonlinear hyperelastic longitudinal plane wave

The perturbation (small-parameter) method is used to analyze the propagation of a harmonic longitudinal plane wave in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model. The three first approximations are obtained, and the contribution of each of them into the wave pattern is analyzed. It is shown that the third approximation somewhat improves the prediction of the evolution of the initial waveprofile: the tendency to generate the second harmonic goes over into the tendency to generate the fourth harmonic Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 46–58, February 2009.     

Analysis of nonlinear oscillations by wavelet transform: Lyapunov exponents

In this paper we give the definition of exponents which would look like Lyapunov exponents in the cases of non-smooth flows of differential equations or iterated maps, and carry back Lyapunov exponents in smooth cases. Here we test our definition by using some simple linear and nonlinear smooth examples.     

Problem of the free oscillations of a capillary bridge

The free oscillations of a capillary bridge whose equilibrium shape is determined by the surface tension forces and the static gravity field are investigated. The values of 25 “lower” levels of the spectrum of natural oscillations of the capillary bridge are found for various control parameters in accordance with the experimental conditions.     

Transverse Vibrations of a Centrally Clamped Rotating Circular Disk

The transverse vibrations of a circular disk of uniform thickness rotatingabout its axis with constant angular velocity are analyzed. The resultsspecialized to the linear case of disks clamped at the center and free atthe periphery are in good agreement with those reported in the literature.The natural frequencies of spinning hard and floppy disks are obtained for various nodal diameters and nodal circles. Primary resonance is shown to occur at the critical rotational speed at which, in the linear analysis, the spinning disk is unable to support arbitraryspatially fixed transverse loads. Using the method of multiple scales, wedetermine a set of four nonlinear ordinary-differential equations governingthe modulation of the amplitudes and phases of two interacting modes. Thesymmetry of the system and the loading conditions are reflected in thesymmetry of the modulation equations. They are reduced to an equivalentset of two first-order equations whose equilibrium solutions aredetermined analytically. The stability characteristics of thesesolutions is studied; the qualitative behavior of the response isindependent of the mode being considered.     

Optimal control of quadratic functionals for affine nonlinear systems

In this paper we analyze the optimal control problem for a class of affine nonlinear systems under the assumption that the associated Lie algebra is nilpotent. The Lie brackets generated by the vector fields which define the nonlinear system represent a remarkable mathematical instrument for the control of affine systems. We determine the optimal control which corresponds to the nilpotent operator of the first order. In particular, we obtain the control that minimizes the energy of the given nonlinear system. Applications of this control to bilinear systems with first order nilpotent operator are considered.     

Piecewise constrained optimization harmonic balance method for predicting the limit cycle oscillations of an airfoil with various nonlinear structures

A method is proposed to calculate the periodic solutions of piecewise nonlinear systems. The method is based on analytical derivation of nonlinear multi-harmonic equations of motion. Since periodic variations of nonlinear forces are characterized by different states, the vibration cycle is broken into sequential transition intervals according to the instant sets of state transitions. Analytical formulations of the harmonic coefficients of the nonlinear forces and its derivatives with respect to the harmonic coefficients of displacements are developed. Sensitivities of the harmonic coefficients of periodic solutions are determined for constructing explicit expressions for vibration amplitude levels as a function of structural parameters. Numerical investigations of the limit cycle oscillations and its sensitivities of an airfoil with different piecewise nonlinearities have been performed. The results show that the developed method is capable of determining the periodic solutions and its sensitivities with respect to the structural parameters. In order to guarantee time continuity of the nonlinear force, for the hysteresis model it is not right to track the periodic solutions by using the preload or freeplay as the continuation parameters.     

Large flexural vibrations of thermally stressed layered shallow shells

A dynamic nonlinear theory for layered shallow shells is derived by means of the von Karman-Tsien theory, modified by the generalized Berger-approximation. Moderately thick shells with polygonal planform composed of multiple perfectly bonded layers are considered. The shell edges are assumed to be prevented from in-plane motions and are simply supported. A distributed lateral force loading is applied to the structure, and additionally, the influence of a static thermal prestress, corresponding to a spatial distribution of cross-sectional mean temperature, is taken into account. In the special case of laminated shells made of transversely isotropic layers with physical properties symmetrically distributed about the middle surface, a correspondence to moderately thick homogeneous shells is found. Application of a multi-mode expansion in the Galerkin procedure to the governing differential equation, where the eigenfunctions of the corresponding linear plate problem are used as space variables, renders a coupled set of ordinary time differential equations for the generalized coordinates with cubic as well as quadratic nonlinearities. The nonlinear steady-state response of shallow shells subjected to a time-harmonic lateral excitation is investigated and the phenomenon of primary resonance is studied by means of the perturbation method of multiple scales. A unifying non-dimensional representation of the nonlinear frequency response function is presented that is independent of the special shell planform.     

The existence of periodic solutions for nonlinear systems of first-order differential equations at resonance

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On the self-switching of hypersonic waves in cubic nonlinear elastic nanocomposites

A summary on transistors and some facts on nanocomposite materials and their classical models are provided. New models used here for computer simulation are described. Results from a theoretical study of the interaction of cubic nonlinear harmonic elastic plane waves in a Murnaghan material are presented. The interaction of two harmonic waves is analyzed using the method of slowly varying amplitudes. The mechanism of energy pumping from a strong pump wave to a weak signal wave is examined. The theoretical and numerical analyses conducted suggest that in theory, a nanocomposite material may be used to create a transistor that would work with hypersonic waves and have a speed in the nanosecond range Translated from Prikladnaya Mekhanika, Vol. 45, No. 1, pp. 90–117, January 2009.     

一类多自由度强非线性振动系统主共振的渐近解法

提出一种渐近方法用来处理一类多自由度强非线性自治振动系统，它是新渐近方法￣［１］的推广。本方法适用于主共振情形，我们建立了振幅和相位所满足的方程。文末用两个例子说明本方法的有效性。