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TECHNICAL PAPERS

Vibration Control Using Parametric Excitation

[+] Author and Article Information
Phillip H. Nguyen, Jerry H. Ginsberg

The Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405

J. Vib. Acoust 123(3), 359-364 (Feb 01, 2001) (6 pages) doi:10.1115/1.1377019 History: Received April 01, 2000; Revised February 01, 2001
Copyright © 2001 by ASME
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References

Hirsch,  P., 1930, “Das Pendel mit oszillierendem Aufhangepunkt,” Z. Angew. Math. Mech., 10, pp. 41–52.
Sethna, P. R., and Hemp, G. W., 1965, “Nonlinear Oscillations of a Gyroscopic Pendulum with an Oscillating Point of Suspension,” Les Vibrations Forcées dans les Systèmes Non-Linéaires, Center National de la Recherche Scientifique, Paris, pp. 375–391.
Ness,  D. J., 1967, “Small Oscillations of a Stabilized, Inverted Pendulum,” Am. J. Phys., 35, pp. 964–967.
Meirovitch, L., 1970, Methods of Analytical Dynamics, McGraw Hill, New York.
Troger,  H., 1975, “Bemerkungen zum Pendel mit oszillierendem Aufhangepunkt,” Z. Angew. Math. Mech., 55, pp. 68–69.
Jordan, D. W., and Smith, P., 1987, Nonlinear Ordinary Differential Equations, Clarendon Press, Oxford.
Skalak,  R., and Yarymovych,  M. I., 1960, “Subharmonic Oscillations of a Pendulum,” ASME J. Appl. Mech., 27, pp. 159–164.
Struble,  R. A., 1963, “On the Subharmonic Oscillations of a Pendulum,” ASME J. Appl. Mech., 30, pp. 301–303.
Dugundji,  J., and Chhatpar,  C. K., 1970, “Dynamic Stability of a Pendulum Under Parametric Excitation,” Rev. Roum. Sci. Tech., Ser. Mec. Appl., 15, No. 4, pp. 741–763.
Chester,  W., 1975, “The Forced Oscillations of a Simple Pendulum,” J. Inst. Math. Appl., 15, pp. 289–306.
Cartmell, M., 1990, Introduction to Linear, Parametric and Nonlinear Vibrations, Chapman and Hall, London, New York, Tokyo, Melbourne, Madras.
Frolov, K. V., 1967, “Parametric and Autoparametric Oscillations of some Nonlinear Mechanical Systems,” Proceedings of the 4th Conference on Nonlinear Oscillations, pp. 327–336.
Hsu,  C. S., and Cheng,  W.-H., 1974, “Steady-State Response of a Dynamical System Under Combined Parametric and Forcing Excitations,” ASME J. Appl. Mech., 41, pp. 371–378.
Ness,  D. J., 1971, “Resonance Classification in a Cubic System,” ASME J. Appl. Mech., 38, pp. 585–590.
Nguyen,  D. V., 1975, “Interaction Between Parametric and Forced Oscillations in Multidimensional Systems,” J. Tech. Phys., 16, pp. 213–225.
Troger,  H., and Hsu,  C. S., 1977, “Response of a Nonlinear System Under Combined Parametric and Forcing Excitation,” ASME J. Appl. Mech., 44, pp. 179–181.
HaQuang,  N., Mook,  D. T., and Plaut,  R. H., 1987, “Non-Linear Structural Vibrations Under Combined Parametric and External Excitations,” J. Sound Vib., 118, No. 2, pp. 291–306.
HaQuang,  N., Mook,  D. T., and Plaut,  R. H., 1987, “A Non-Linear Analysis of the Interactions Between Parametric and External Excitations,” J. Sound Vib., 118, No. 3, pp. 425–439.
Plaut,  R. H., Gentry,  J. J., and Mook,  D. T., 1990, “Non-Linear Structural Vibrations Under Combined Multi-Frequency Parametric and External Excitations,” J. Sound Vib., 140, No. 3, pp. 381–390.

Figures

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Pendulum with a vertically translating pivot
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Temporal response for low frequency parametric excitation, r=0.95
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Temporal response for moderate frequency parametric excitation, r=0.95
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Temporal response for high frequency parametric excitation, r=0.95
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Amplitude factor as a function of nondimensional pivot frequency 1/α,r=0.95, ϕ=0
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Amplitude factor as a function of pivot amplitude, r=0.95
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Comparison of numerical transient response and two-term analytical Fourier series, r=0.95, α=0.123, β=0.24, ϕ=0
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Amplitude factor as a function of nondimensional pivot frequency 1/α,r=0.7, ϕ=0
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Pivot frequency and amplitude required to achieve γ≈0.1 at a specified excitation frequency
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Amplitude factor as a function of pivot amplitude, r=2
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Frequency response for different combinations of α and β
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Amplitude factor for different combinations of α and β

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