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  • 1
    Electronic Resource
    Electronic Resource
    Nonlinear motions of beam-mass structure (1990)
    Springer
    Nonlinear dynamics 1 (1990), S. 39-61 
    Details
    ISSN: 1573-269X
    Keywords: structural dynamics ; internal resonance ; modulation equations ; Hopf bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study the planar dynamic response of a flexible L-shaped beam-mass structure with a two-to-one internal resonance to a primary resonance. The structure is subjected to low excitation (mili g) levels and the resulting nonlinear motions are examined. The Lagrangian for weakly nonlinear motions of the undamped structure is formulated and time averaged over the period of the primary oscillation, leading to an autonomous system of equations governing the amplitudes and phases of the modes involved in the internal resonance. Later, modal damping is assumed and modal-damping coefficients, determined from experiments, are included in the analytical model. The locations of the saddle-node and Hopf bifurcations predicted by the analysis are in good agreement, respectively, with the jumps and transitions from periodic to quasi-periodic motions observed in the experiments. The current study is relevant to the dynamics and modeling of other structural systems as well.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01857584
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  • 2
    Electronic Resource
    Electronic Resource
    Coupling between high-frequency modes and a low-frequency mode: Theory and experiment (1996)
    Springer
    Nonlinear dynamics 11 (1996), S. 17-36 
    Details
    ISSN: 1573-269X
    Keywords: Widely separated natural frequencies ; energy transfer ; internal resonance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An analytical and experimental investigation into the response of a nonlinear continuous system with widely separated natural frequencies is presented. The system investigated is a thin, slightly curved, isotropic, flexible cantilever beam mounted vertically. In the experiments, for certain vertical harmonic base excitations, we observed that the response consisted of the first, third, and fourth modes. In these cases, the modulation frequency of the amplitudes and phases of the third and fourth modes was equal to the response frequency of the first mode. Subsequently, we developed an analytical model to explain the interactions between the widely separated modes observed in the experiments. We used a three-mode Galerkin projection of the partial-differential equation governing a thin, isotropic, inextensional beam and obtained a sixth-order nonautonomous system of equations by using an unconventional coordinate transformation. In the analytical model, we used experimentally determined damping coefficients. From this nonautonomous system, we obtained a first approximation of the response by using the method of averaging. The analytically predicted responses and bifurcation diagrams show good qualitative agreement with the experimental observations. The current study brings to light a new type of nonlinear motion not reported before in the literature and should be of relevance to many structural and mechanical systems. In this motion, a static response of a low-frequency mode interacts with the dynamic response of two high-frequency modes. This motion loses stability, resulting in oscillations of the low-frequency mode accompanied by a modulation of the amplitudes and phases of the high-frequency modes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00045049
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  • 3
    Electronic Resource
    Electronic Resource
    Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams (1999)
    Springer
    Nonlinear dynamics 20 (1999), S. 131-158 
    Details
    ISSN: 1573-269X
    Keywords: three-to-one resonance ; internal resonance ; beam vibrations ; bifurcation ; blue-sky catastrophe
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear planar response of a hinged-clamped beam to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of static axial loads, resulting in a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear integral-partial-differential equation and associated boundary conditions and derive three sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the cases of (a) principal parametric resonance of either the first or the second mode, and (b) a combination parametric resonance of the additive type of these modes. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of principal parametric resonance of the first mode or combination parametric resonance of the additive type, trivial and two-mode solutions are possible, whereas for the case of parametric resonance of the second mode, trivial, single, and two-mode solutions are possible. The trivial and two-mode equilibrium solutions of the modulation equations may undergo either a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. For some excitation parameters, we found complex responses including period-doubling bifurcations and blue-sky catastrophes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008310419911
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  • 4
    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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  • 5
    Electronic Resource
    Electronic Resource
    Cyclic motions near a Hopf bifurcation of a four-dimensional system (1992)
    Springer
    Nonlinear dynamics 3 (1992), S. 19-39 
    Details
    ISSN: 1573-269X
    Keywords: Hopf bifurcation ; multiple scales ; limit cycles ; internal resonance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study motions near a Hopf bifurcation of a representative nonconservative four-dimensional autonomous system with quadratic nonlinearities. Special cases of the four-dimensional system represent the envelope equations that govern the amplitudes and phases of the modes of an internally resonant structure subjected to resonant excitations. Using the method of multiple scales, we reduce the Hopf bifurcation problem to two differential equations for the amplitude and phase of the bifurcating cyclic solutions. Constant solutions of these equations provide asymptotic expansions for the frequency and amplitude of the bifurcating limit cycle. The stability of the constant solutions determines the nature of the bifurcation (i.e., subcritical or supercritical). For different choices of the control parameter, the range of validity of the analytical approximation is ascertained using numerical simulations. The perturbation analysis and discussions are also pertinent to other autonomous systems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00045469
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  • 6
    Electronic Resource
    Electronic Resource
    Three-to-One Internal Resonances in Hinged-Clamped Beams (1997)
    Springer
    Nonlinear dynamics 12 (1997), S. 129-154 
    Details
    ISSN: 1573-269X
    Keywords: Three-to-one resonance ; internal resonance ; beam vibrations ; bifurcation ; crises
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear planar response of a hinged-clamped beam to a primary excitation of either its first mode or its second mode is investigated. The analysis accounts for mid-plane stretching, a static axial load and a restraining spring at one end, and modal damping. For a range of axial loads, the second natural frequency is approximately three times the first natural frequency and hence the first and second modes may interact due to a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear partial-differential equation and derive two sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the case of primary resonance of either the first or the second mode. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of primary resonance of the first mode, only two-mode solutions are possible, whereas for the case of primary resonance of the second mode, single- and two-mode solutions are possible. The two-mode equilibrium solutions of the modulation equations may undergo a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. A shooting technique is used to calculate limit cycles of the modulation equations and Floquet theory is used to ascertain their stability. The limit cycles correspond to periodically modulated motions of the beam. The limit cycles are found to undergo cyclic-fold bifurcations and period-doubling bifurcations, leading to chaos. The chaotic attractors may undergo boundary crises, resulting in the destruction of the chaotic attractors and their basins of attraction.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008229503164
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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 Normal Modes of Buckled Beams: Three-to-One and One-to-One Internal Resonances (1999)
    Springer
    Nonlinear dynamics 18 (1999), S. 253-273 
    Details
    ISSN: 1573-269X
    Keywords: Nonlinear normal modes ; internal resonance ; buckled beam ; perturbation methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Nonlinear normal modes of a fixed-fixed buckled beam about its first post-buckling configuration are investigated. The cases of three-to-one and one-to-one internal resonances are analyzed. Approximate solutions for the nonlinear normal modes are computed by applying the method of multiple scales directly to the governing integral-partial-differential equation and associated boundary conditions. Curves displaying variation of the amplitude of one of the modes with the internal-resonance-detuning parameter are generated. It is shown that, for a three-to-one internal resonance between the first and third modes, the beam may possess one stable uncoupled mode (high-frequency mode) and either (a) one stable coupled mode, (b) three stable coupled modes, or (c) two stable and one unstable coupled modes. For the same resonance, the beam possesses one degenerate mode (with a multiplicity of two) and two stable and one unstable coupled modes. On the other hand, for a one-to-one internal resonance between the first and second modes, the beam possesses (a) two stable uncoupled modes and two stable and two unstable coupled modes; (b) one stable and one unstable uncoupled modes and two stable and two unstable coupled modes; and (c) two stable uncoupled and two unstable coupled modes (with a multiplicity of two). For a one-to-one internal resonance between the third and fourth modes, the beam possesses (a) two stable uncoupled modes and four stable coupled modes; (b) one stable and one unstable uncoupled modes and four stable coupled modes; (c) two unstable uncoupled modes and four stable coupled modes; and (d) two stable uncoupled modes and two stable coupled modes (each with a multiplicity of two).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008389024738
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