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  • 1
    Electronic Resource
    Electronic Resource
    Chaos and instability in a power system — Primary resonant case (1990)
    Springer
    Nonlinear dynamics 1 (1990), S. 313-339 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01865278
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  • 2
    Electronic Resource
    Electronic Resource
    Chaos and instability in a power system: Subharmonic-resonant case (1991)
    Springer
    Nonlinear dynamics 2 (1991), S. 53-72 
    Details
    ISSN: 1573-269X
    Keywords: Power systems ; loss of synchronism ; chaos ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The response of a single-machine quasi-infinite busbar system to the simultaneous occurrence of principal parametric resonance and subharmonic resonance of order one-half is investigated. By numerical simulations we show the existence of oscillatory solutions (limit cycles), period-doubling bifurcations, chaos, and unbounded motions (loss of synchronism). The method of multiple scales is used to derive a second-order analytical solution that predicts (a) the onset of period-doubling bifurcations, which is a precursor to chaos and unbounded motions (loss of synchronism), and (b) saddle-node bifurcations, which may be precursors to loss of synchronism.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00045055
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  • 3
    Electronic Resource
    Electronic Resource
    Nonlinear response of a parametrically excited buckled beam (1993)
    Springer
    Nonlinear dynamics 4 (1993), S. 499-525 
    Details
    ISSN: 1573-269X
    Keywords: Chaos ; buckled beam ; parametric resonance ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A nonlinear analysis of the response of a simply-supported buckled beam to a harmonic axial load is presented. The method of multiple scales is used to determine to second order the amplitude- and phase-modulation equations. Floquet theory is used to analyze the stability of periodic responses. The perturbation results are verified by integrating the governing equation using both digital and analog computers. For small excitation amplitudes, the analytical results are in good agreement with the numerical solutions. The large-amplitude responses are investigated by using a digital computer and are compared with those obtained via an analog-computer simulation. The complicated dynamic behaviors that were found include period-multiplying and period-demultiplying bifurcations, period-three and period-six motions, jump phenomena, and chaos. In some cases, multiple periodic attractors coexist, and a chaotic attractor coexists with a periodic attractor. Phase portraits, spectra of the responses, and a bifurcation set of the many solutions are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00053693
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  • 4
    Electronic Resource
    Electronic Resource
    Observations of modal interactions in resonantly forced beam-mass structures (1991)
    Springer
    Nonlinear dynamics 2 (1991), S. 77-117 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00053831
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  • 5
    Electronic Resource
    Electronic Resource
    Nonlinear random coupled motions of structural elements with quadratic nonlinearities (1991)
    Springer
    Nonlinear dynamics 2 (1991), S. 305-316 
    Details
    ISSN: 1573-269X
    Keywords: Random vibrations ; closure scheme ; autoparametric resonance ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The second-order closure method is used to analyze the nonlinear response of two-degree-of-freedom systems with quadratic nonlinearities. The excitation is assumed to be the sum of a deterministic harmonic component and a random component. The case of primary resonance of the second mode in the presence of a two-to-one internal (autoparametric) resonance is investigated. The method of multiple scales is used to obtain four first-order ordinary-differential equations that describe the modulation of the amplitudes and phases of the two modes. Applying the second-order closure method to the modulation equations, we determine the stationary mean and mean-square responses. For the case of a narrow-band random excitation, the results show that the presence of the nonlinearity causes multi-valued mean-square responses. The multi-valuedness is responsible for a jump phenomenon. Contrary to the results of the linear analysis, the nonlinear analysis reveals that the directly excited second mode takes a small amount of the input energy (saturates) and spills over the rest of the input energy into the first mode, which is indirectly excited through the autoparametric resonance.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00045299
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  • 6
    Electronic Resource
    Electronic Resource
    Dynamics of a Cubic Nonlinear Vibration Absorber (1999)
    Springer
    Nonlinear dynamics 20 (1999), S. 283-295 
    Details
    ISSN: 1573-269X
    Keywords: vibration absorber ; saturation ; internal resonance ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study the dynamics of a nonlinear active vibration absorber. We consider a plant model possessing curvature and inertia nonlinearities and introduce a second-order absorber that is coupled with the plant through user-defined cubic nonlinearities. When the plant is excited at primary resonance and the absorber frequency is approximately equal to the plant natural frequency, we show the existence of a saturation phenomenon. As the forcing amplitude is increased beyond a certain threshold, the response amplitude of the directly excited mode (plant) remains constant, while the response amplitude of the indirectly excited mode (absorber) increases. We obtain an approximate solution to the governing equations using the method of multiple scales and show that the system possesses two possible saturation values. Using numerical techniques, we perform stability analyses and demonstrate that the system exhibits complicated dynamics, such as Hopf bifurcations, intermittency, and chaotic responses.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008358825502
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  • 7
    Electronic Resource
    Electronic Resource
    Nonlinear Nonplanar Dynamics of Parametrically Excited Cantilever Beams (1998)
    Springer
    Nonlinear dynamics 15 (1998), S. 31-61 
    Details
    ISSN: 1573-269X
    Keywords: Beams ; internal resonance ; parametric resonance ; bifurcations ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its flexural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to two integro-partial-differential equations governing the motions of the beams. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. These modulation equations exhibit symmetry properties. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, Hopf, and codimension-2 bifurcations. A detailed bifurcation analysis of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008218009139
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  • 8
    Electronic Resource
    Electronic Resource
    Experimental Investigation of Single-Mode Responses in a Fixed-Fixed Buckled Beam (1998)
    Springer
    Nonlinear dynamics 15 (1998), S. 155-177 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008231012968
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  • 9
    Electronic Resource
    Electronic Resource
    Experimental Validation of Reduction Methods for Nonlinear Vibrations of Distributed-Parameter Systems: Analysis of a Buckled Beam (1998)
    Springer
    Nonlinear dynamics 17 (1998), S. 95-117 
    Details
    ISSN: 1573-269X
    Keywords: Buckled beam ; experiment ; Galerkin method ; direct approach ; method of multiple scales
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An experimental validation of the suitability of reduction methods for studying nonlinear vibrations of distributed-parameter systems is attempted. Nonlinear planar vibrations of a clamped-clamped buckled beam about its first post-buckling configuration are analyzed. The case of primary resonance of the nth mode of the beam, when no internal resonances involving this mode are active, is investigated. Approximate solutions are obtained by applying the method of multiple scales to a single-mode model discretized via the Galerkin procedure and by directly attacking the governing integro-partial-differential equation and boundary conditions with the method of multiple scales. Frequency-response curves for the case of primary resonance of the first mode are generated using both approaches for several buckling levels and are contrasted with experimentally obtained frequency-response curves for two test beams. For high buckling levels above the first crossover point of the beam, the computed frequency-response curves are qualitatively as well as quantitatively different. The experimentally obtained frequency-response curves for the directly excited first mode are in agreement with those obtained with the direct approach and in disagreement with those obtained with the single-mode discretization approach.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008389810246
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  • 10
    Electronic Resource
    Electronic Resource
    Nonlinear interactions in a parametrically excited system with widely spaced frequencies (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 195-216 
    Details
    ISSN: 1573-269X
    Keywords: Energy transfer ; bifurcations ; chaos ; crises
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An investigation is presented into the transfer of energy from high- to low-frequency modes. The method of averaging is used to analyze the response of a two-degree-of-freedom system with widely spaced frequencies and cubic nonlinearities to a principal parametric resonance of the high-frequency mode. The conditions under which energy can be transferred from high- to low-frequency modes, as observed in the experiments, are determined. The interactions between the widely separated modes result in various bifurcations, the coexistence of multiple attractors, and chaotic attractors. The results show that damping may be destabilizing. The analytical results are validated by numerically solving the original system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00053708
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