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  • 1
    Electronic Resource
    Electronic Resource
    A theoretical and experimental investigation of a three-degree-of-freedom structure (1994)
    Springer
    Nonlinear dynamics 6 (1994), S. 353-374 
    Details
    ISSN: 1573-269X
    Keywords: Autoparametric resonance ; combination resonance ; saturation ; phase-locked motions ; quasiperiodic motions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The response of a structure to a simple-harmonic excitation is investigated theoretically and experimentally. The structure consists of two light-weight beams arranged in a T-shape turned on its side. Relatively heavy and concentrated weights are placed at the upper and lower free ends and at the point where the two beams are joined. The base of the ‘T’ is clamped to the head of a shaker. Because the masses of the concentrated weights are much larger than the masses of the beams, the first three natural frequencies are far below the fourth; consequently, for relatively low frequencies of the excitation, the structure has, for all practical purposes, only three degrees of freedom. The lengths and weights are chosen so that the third natural frequency is approximately equal to the sum of the two lower natural frequencies, an arrangement that produces an autoparametric (also called an internal) resonance. A linear analysis is performed to predict the natural frequencies and to aid in the design of the experiment; the predictions and observations are in close agreement. Then a nonlinear analysis of the response to a prescribed transverse motion at the base of the ‘T’ is performed. The method of multiple scales is used to obtain six first-order differential equations describing the modulations of the amplitudes and phases of the three interacting modes when the frequency of the excitation is near the third natural frequency. Some of the predicted phenomena include periodic, two-period quasiperiodic, and phase-locked (also called synchronized) motions; coexistence of multiple stable motions and the attendant jumps; and saturation. All the predictions are confirmed in the experiments, and some phenomena that are not yet explained by theory are observed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00053391
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  • 2
    Electronic Resource
    Electronic Resource
    On methods for continuous systems with quadratic and cubic nonlinearities (1992)
    Springer
    Nonlinear dynamics 3 (1992), S. 145-162 
    Details
    ISSN: 1573-269X
    Keywords: Averaged Lagrangian ; valve ; resonances ; Galerkin procedure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Methods for determining the response of continuous systems with quadratic and cubic nonlinearities are discussed. We show by means of a simple example that perturbation and computational methods based on first discretizing the systems may lead to erroncous results whereas perturbation methods that attack directly the nonlinear partial-differential equations and boundary conditions avoid the pitfalls associated with the analysis of the discretized systems. We describe a perturbation technique that applies either the method of multiple scales or the method of averaging to the Lagrangian of the system rather than the partial-differential equations and boundary conditions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00118990
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  • 3
    Electronic Resource
    Electronic Resource
    Whirling of a forced cantilevered beam with static deflection. I: Primary resonance (1993)
    Springer
    Nonlinear dynamics 4 (1993), S. 227-249 
    Details
    ISSN: 1573-269X
    Keywords: Cantilevered beams ; nonlinear analysis ; whirling motion ; primary resonances
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The response of a slender, clastic, cantilevered beam to a transverse, vertical, harmonic excitation is investigated. The effects of nonlinear curvature, nonlinear inertia, viscous damping and static load are included. Previous work often has neglected the static deflection caused by the weight of the beam, which adds quadratic terms in the governing equations of motion. Galerkin's method is used with three modes and approximate solutions of the temporal equations are obtained by the method of multiple scales. Primary resonance is treated here, and out-of-plane motion is possible in the first and second modes when the principal moments of inertia of the beam cross-section are approximately equal. In Parts II and III, secondary resonances and nonstationary passages through various resonances are considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00046322
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  • 4
    Electronic Resource
    Electronic Resource
    Whirling of a forced cantilevered beam with static deflection. II: Superharmonic and subharmonic resonances (1993)
    Springer
    Nonlinear dynamics 4 (1993), S. 337-356 
    Details
    ISSN: 1573-269X
    Keywords: Cantilevered beams ; nonlinear analysis ; whirling motion ; secondary resonances
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Secondary resonances of a slender, elastic, cantilevered beam subjected to a transverse harmonic load are investigated. The effects of nonlinear curvature, nonlinear inertia, viscous damping and static load are included. Cubic terms in the governing equations lead to subharmonic and superharmonic resonances of order three. The static displacement produced by the weight of the beam introduces quadratic terms in the governing equations, which cause subharmonic and superharmonic resonances of order two. Out-of-plane motion is possible in all of these secondary resonances when the principal moments of inertia of the beam cross section are approximately equal.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00120670
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  • 5
    Electronic Resource
    Electronic Resource
    Whirling of a forced cantilevered beam with static deflection. III: Passage through resonance (1993)
    Springer
    Nonlinear dynamics 4 (1993), S. 461-481 
    Details
    ISSN: 1573-269X
    Keywords: Cantilevered beams ; nonlinear analysis ; transient responses ; passage through resonance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Nonstationary excitations of slender, elastic, cantilevered beams with equal principal moments of inertia are considered. The excitation frequency is slowly increased or decreased through a resonance of the first mode at a constant rate. Three resonances are investigated: primary resonance, superharmonic resonance of order two and subharmonic resonance of order two. After application of Galerkin's method with three modes, the nonlinear, nonstationary response of the first mode of the beam is determined by two methods: integration of the modulation equations obtained from the method of multiple scales, and direct numerical integration of the temporal equations of motion. Time histories are presented and the effects of excitation amplitude, rate of acceleration or deceleration through resonance, damping and initial conditions of the disturbance on the maximum response are studied. The effect of a persistent random disturbance is also examined. Although the excitation acts in the vertical plane, whirling occurs if the beam is subjected to out-of-plane disturbances.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00053691
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  • 6
    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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  • 7
    Electronic Resource
    Electronic Resource
    Combination resonances in the response of the duffing oscillator to a three-frequency excitation (1998)
    Springer
    Acta mechanica 131 (1998), S. 235-245 
    Details
    ISSN: 1619-6937
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Summary The response of a weakly nonlinear, single-degree-of-freedom system with cubic nonlinearities to multifrequency excitations is studied analytically and numerically. The method of multiple scales is used to obtain uniformly valid, approximate solutions of the governing equation for various combination resonances. The analytical and numerical solutions are in virtually perfect agreement for all cases considered, but difer markedly from the exact solution of the linearized equation of motion. The peak amplitudes in the solutions of the nonlinear equation can be several times those in the solutions of the linearized equation, and they can occur rather more often; moreover, the addition of a static load can affect the natural frequency and, hence, either tune or detune a resonance, producing profound changes in the response. The present results demonstrate that the actual response of a structure can lead to a fatigue life that is much shorter than what is predicted by linear analysis. Hence, the conventional structural engineering practice of considering the structure to be safe from resonant responses when none of the frequencies of the excitation matches the natural frequency is shown to be fraught with danger; a practicing engineer, therefore, cannot afford to be ignorant of nonlinear phenomena.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01177227
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  • 8
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    Combination resonances in the response of the duffing oscillator to a three-frequency excitation (1998)
    Springer
    Details
    Publication Date: 1998-09-01
    Print ISSN: 0001-5970
    Electronic ISSN: 1619-6937
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Published by Springer
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  • 9
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    On methods for continuous systems with quadratic and cubic nonlinearities (1992)
    Springer
    Details
    Publication Date: 1992-01-01
    Print ISSN: 0924-090X
    Electronic ISSN: 1573-269X
    Topics: Mathematics
    Published by Springer
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  • 10
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    Cargo Pendulation Reduction of Ship-Mounted Cranes (2004)
    Springer
    Details
    Publication Date: 2004-02-01
    Print ISSN: 0924-090X
    Electronic ISSN: 1573-269X
    Topics: Mathematics
    Published by Springer
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