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  • 1
    ISSN: 1573-269X
    Keywords: numerical simulation ; chaos ; cable ; resonances ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The finite motions of a suspended elastic cable subjected to a planar harmonic excitation can be studied accurately enough through a single ordinary-differential equation with quadratic and cubic nonlinearities. The possible onset of chaotic motion for the cable in the region between the one-half subharmonic resonance condition and the primary one is analysed via numerical simulations. Chaotic charts in the parameter space of the excitation are obtained and the transition from periodic to chaotic regimes is analysed in detail by using phase-plane portraits, Poincaré maps, frequency-power spectra, Lyapunov exponents and fractal dimensions as chaotic measures. Period-doubling, sudden changes and intermittency bifurcations are observed.
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  • 2
    ISSN: 1573-269X
    Keywords: Power systems ; chaos ; bifurcations ; loss of synchronism
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We investigate some of the instabilities in a single-machine quasi-infinite busbar system. The system's behavior is described by the so-called swing equation, which is a nonlinear second-order ordinary-differential equation with additive and multiplicative harmonic terms having the frequency Ω. When Ω≈ω0, where ω0 is the linear natural frequency of the machine, we use digital-computer simulations to exhibit some of the complicated responses of the machine, including period-doubling bifurcations, chaotic motions, and unbounded motions (loss of synchronism). To predict the onset of these complicated behaviors, we use the method of multiple scales to develop an approximate first-order closed-form expression for the period-one responses of the machine. Then, we use various techniques to determine the stability of the analytical solutions. The analytically predicted period-one solutions and conditions for its instability are in good agreement with the digital-computer results.
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 10 (1996), S. 31-48 
    ISSN: 1573-269X
    Keywords: Experimental ; nonstationary ; bifurcations ; vibrations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper considers the resonant behavior of a mechanical oscillator during a linear frequency sweep. Both numerical and experimental results are presented. The experimental system consisting of a track in the shape of a potential energy surfaces has been used to highlight other types of nonlinear behavior and is here adapted so that the forcing frequency can be evolved continuously in time. The classic linear oscillator (with a parabolic potential well) is used as an introduction to illustrate basic features of the experiment and its response. Then, a track with a double well is used to assess nonstationary frequency effects on certain nonlinear characteristics, specifically amplitude jumps and flip bifurcations.
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  • 4
    ISSN: 1573-269X
    Keywords: Chaos ; numerical simulation ; spacecraft ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The dynamics of a simplified model of a spinning spacecraft with a circumferential nutational damper is investigated using numerical simulations for nonlinear phenomena. A realistic spacecraft parameter configuration is investigated and is found to exhibit chaotic motion when a sinusoidally varying torque is applied to the spacecraft for a range of forcing amplitude and frequency. Such a torque, in practice, may arise in the platform of a dual-spin spacecraft under malfunction of the control system or from an unbalanced rotor or from vibrations in appendages. The equations of motion of the model are derived with Lagrange's equations using a generalisation of the kinetic energy equation and a linear stability analysis is given. Numerical simulations for satellite parameters are performed and the system is found to exhibit chaotic motion when a sinusoidally varying torque is applied to the spacecraft for a range of forcing amplitude and frequency. The motion is studied by means of time history, phase space, frequency spectrum, Poincaré map, Lyapunov characteristic exponents and Correlation Dimension. For sufficiently large values of torque amplitude, the behaviour of the system was found to have much in common with a two well potential problem such as a Duffing oscillator. Evidence is also presented, indicating that the onset of chaotic motion was characterised by period doubling as well as intermittency.
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  • 5
    ISSN: 1573-269X
    Keywords: Buckled beam ; clamp design ; asymmetric responses ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear, single-mode responses of a fixed-fixed, buckled beam are investigated under the case of a uniform, transverse, harmonic excitation. In order to avoid axial slipping and to obtain meaningful data, a clamping apparatus was designed to maximize the clamping force applied to the beam. To fully characterize the single-mode responses, data were obtained at various levels of buckling up to 3.3 times the thickness of the beam. The data demonstrate that at a low level of buckling, supercritical period doubling occurs during an amplitude sweep in which the first mode is directly excited. However, as the buckling level increases, the period-doubling bifurcation becomes subcritical during such amplitude sweeps. In addition, a period-five motion, broadband responses, and responses with an unexplained sideband structure were observed.
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  • 6
    ISSN: 1573-269X
    Keywords: Internal resonances ; bifurcations ; quasiperiodic motions ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We present a collection of experimental results on the influence of modal interactions (i.e., internal or autoparametric resonances) on the nonlinear response of flexible metallic and composite structures subjected to a range of resonant excitations. The experimental results are provided in the form of frequency spectra, Poincaré sections, pseudo-phase planes, dimension calculations, and response curves. Experimental observations of transitions from periodic to chaotically modulated motions are also presented. We also discuss relevant analytical results. The current study is also relevant to other internally resonant structural systems.
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  • 7
    ISSN: 1573-269X
    Keywords: Railway dynamics ; limit-cycles ; bifurcations ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A dicone moving on a pair of cylindrical rails can be considered as a simplified model of a railway wheelset. Taking into account the non-linear friction laws of rolling contact, the equations of motion for this non-linear mechanical system result in a set of differential-algebraic equations. Previous simulations performed with the differential-algebraic solver DASSL, [2], and experiments, [7], indicated non-linear phenomena such as limit-cycles, bifurcations as well as chaotic behaviour. In this paper the non-linear phenomena are investigated in more detail with the aid of special in-house software and the path-following algorithm PATH [10]. We apply Poincaré sections and Poincaré maps to describe the structure of periodic, quasiperiodic and chaotic motions. The analyses show that part of the chaotic behaviour of the non-linear system can be fully understood as a non-linear iterative process. The resulting stretching and folding processes are illustrated by series of Poincaré sections.
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  • 8
    ISSN: 1573-269X
    Keywords: bifurcations ; cuspidal loop ; numerical continuation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we develop numerical algorithms for thecontinuation of degenerate homoclinic orbits to non-hyperbolicequilibria in planar systems. The first situation corresponds to asaddle-node equilibrium (a zero eigenvalue) and the second one is theso-called cuspidal loop (double-zero eigenvalue). The methods proposedmay deal with codimension-two and -three homoclinic connections.Application of the algorithms to several examples supports its validityand demonstrates its usefulness.
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 6 (1994), S. 285-299 
    ISSN: 1573-269X
    Keywords: Impact oscillators ; bifurcations ; grazing
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Models of impact oscillators using an instantaneous impact law are by their very nature discontinuous. These discontinuities geve rise to bifurcations which cannot be classified using the usual tools of bifurcation analysis. However, we present numerical evidence which suggests that these discontinuous bifurcations are just the limits (in some sense) of standard bifurcations of smooth dynamical systems as the impact is hardened. Finally we show how one dimensional maps of the interval with essentially similar characteristics can exhibit the same kinds of bifurcational behaviour, and how these bifurcations are related to standard bifurcations.
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  • 10
    ISSN: 1573-269X
    Keywords: Experimental analysis ; cables ; nonlinear vibrations ; modal interaction ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An experimental model of an elastic cable carrying eight concentrated masses and hanging at in-phase or out-of-phase vertically moving supports is considered. The system parameters are adjusted to approximately realize multiple 1:1 and 2:1 internal resonance conditions involving planar and nonplanar, symmetric and antisymmetric modes. Response measurements are made in various frequency ranges including meaningful external resonance conditions. A ‘local’ analysis of the system response is made on the basis of numerous amplitude-frequency and amplitude-forcing plots obtained in different ranges of the control parameter space. Attention is mainly devoted to the detection of the main features of the regular motions exhibited by the system, and to the analysis of the relevant phenomena of nonlinear modal interaction, competition, and local bifurcation between planar and nonplanar regular responses. The resulting picture appears very rich and varied.
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