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TECHNICAL PAPERS

Discrete Frequency Models: A New Approach to Temporal Analysis

[+] Author and Article Information
Douglas E. Adams

School of Mechanical Engineering, Purdue University, 1077 Ray W. Herrick Laboratories, West Lafayette, IN 47907-1077

Randall J. Allemang

Structural Dynamics Research Laboratory (UC-SDRL), University of Cincinnati, Cincinnati, OH 45221-0072

J. Vib. Acoust 123(1), 98-103 (Aug 01, 2000) (6 pages) doi:10.1115/1.1320815 History: Received November 01, 1999; Revised August 01, 2000
Copyright © 2001 by ASME
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References

Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, Wiley, New York.
Beranek, L. L., 1993, Acoustics, Acoustical Society of America, Woodbury, New York.
Schetzen, M., 1989, The Volterra and Wiener Theories of Nonlinear Systems, Krieger Publishing, Malabar.
Leontaritis,  I. J., and Billings,  S. A., 1985, “Input-Output Parametric Models for Nonlinear Systems Part I: Deterministic Nonlinear Systems,” Int. J. Control, 41, pp. 303–328.
Chen,  S., and Billings,  S. A., 1989, “Representations of Nonlinear Systems: The NARMAX Model,” Int. J. Control, 49, pp. 1012–1032.
Storer,  D. M., and Tomlinson,  G. R., 1993, “Recent Developments in the Measurement and Interpretation of Higher Order Transfer Functions from Non-Linear Structures,” Mech. Syst. Signal Process., 7, pp. 173–189.
Storer, D. M., and Tomlinson, G. R., 1991, “An Explanation of the Cause of the Distortion in the Transfer Function of a Duffing Oscillator Subject to Sine Excitation,” Proceedings of the International Modal Analysis Conference, Vol. 2, pp. 1197–1205.
Collis,  W. B., White,  P. R., and Hammond,  J. K., 1998, “Higher-Order Spectra: The Bispectrum and Trispectrum,” Mech. Syst. Signal Process., 12, pp. 375–394.
Bendat, J. S., 1998, “Nonlinear Systems Techniques and Applications,” Wiley Sons, New York.
Mohammad,  K. S., Worden,  K., and Tomlinson,  G. R., 1992, “Direct Parameter Estimation for Linear and Non-Linear Structures,” Sound Vib., 152, pp. 471–499.
Ljung, L., 1987, System Identification Theory for the User., PTR Prentice Hall, Englewood Cliffs, New Jersey.
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Strogatz, S. H., 1994, Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, Massachusetts.
Adams,  D. E., and Allemang,  R. J., 1999, “Characterization of Nonlinear Vibrating Systems Using Internal Feedback and Frequency Response Modulation,” ASME J. Vibr. Acoust., 121, pp. 495–500.
Adams,  D. E., and Allemang,  R. J., 1999, “A Frequency Domain Method for Estimating the Parameters of a Nonlinear Structural Dynamic Model through Feedback,” Mech. Syst. Signal Process., 14, pp. 637–656.
Vold, H., Crowley, J., and Rocklin, G., 1985, “A Comparison of H1,H2,Hv Frequency Response Functions,” Proceedings of the International Modal Analysis Conference, Vol. 1, pp. 272–278.
Adams, D. E., and Allemang, R. J., 2000, “Residual Frequency Autocorrelation As An Indicator of Nonlinearity,” accepted for publication in Int. J. Non-Linear Mech., in press.
Ayyub, B. M. and McCuen, R. H., 1996, Numerical Methods for Engineers, Prentice-Hall, Upper Saddle River, New Jersey.

Figures

Grahic Jump Location
Forced response spectrum of the system in Eq. (1) to a sinusoidal excitation at 1.5Ω=1.5k/m and 1.2 N amplitude: (—) True response spectrum, (○○○) DFM estimates from Eq. (17).
Grahic Jump Location
Estimates of the frequency response function of the underlying system in Eq. (1) and the residual autocorrelation of these estimates: (Left) Linear system estimate and residual autocorrelation, μ=0(1/m2), (Right) Nonlinear system estimate and residual autocorrelation, μ=7e5(1/m2).
Grahic Jump Location
An illustration of the three fundamental model types and the relationships between them. The discrete frequency model represents a new temporal analysis approach.
Grahic Jump Location
Venn diagram illustration of the variation in linear and nonlinear dynamics with environmental test conditions
Grahic Jump Location
Illustration of nonlinear feedback within a duffing oscillator. Two forces act on the underlying linear system: the external force and a cubic nonlinear function of the response.
Grahic Jump Location
Forced response of the system in Eq. (1) to three sinusoidal excitations with unity amplitudes and different frequencies: (Top) ω0=Ω/2=k/4m, (Middle) ω0=Ω, (Bottom) ω0=2Ω.
Grahic Jump Location
Forced response of the system in Eq. (1) to two sinusoidal excitations at approximately three times the underlying resonant frequency: (Top) F0=1 N,ω0=3.3Ω=3.3k/m, (Bottom) F0=5 N,ω0=3.3Ω where Ω=3.9 Hz.
Grahic Jump Location
Forced response spectrum of the system in Eq. (1) to a sinusoidal excitation at 1.5Ω=1.5k/m and 0.8 N amplitude: (—) True response spectrum, (○○○) DFM estimates from Eq. (13).

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