Global and Local Nonlinear System Responses under Narrowband Random Excitations. I: Semianalytical Method
Publication: Journal of Engineering Mechanics
Volume 133, Issue 1
Abstract
A single-degree-of-freedom nonlinear structural system under narrowband random excitation can exhibit very complex global and local response behaviors. In order to develop a stochastic method to analyze the nonlinear responses, the system under deterministic excitation is first modeled and examined in the primary and subharmonic resonance regions. Typical response behaviors including coexistence of attractors and (global) jump phenomena are observed. Governing equations of the probability for the response–amplitude perturbations (a local transition) within an attraction domain and a jump between different attraction domains (a global transition) are derived under the assumption of a stationary Markov condition. The overall response–amplitude probability distribution is obtained by applying the Bayes formula to the two types of response transition probability distributions. In this study, we focus on understanding the physics of the transitions using the proposed probability method.
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Acknowledgments
Financial support from the United States Office of Naval Research (Grant Nos. ONRN00014-92-J-1221 and ONRN00014-04-10008) is gratefully acknowledged.
References
Bouleau, N., and Lépingle, D. (1994). Numerical methods for stochastic processes, Wiley, New York.
Crandall, S. H., and Mark, W. D. (1963). Random vibration in mechanical systems, Academic, New York.
Davies, H. G., and Liu, Q. (1990). “The response envelope probability density function of a dulling oscillator with random narrow-band excitation.” J. Sound Vib., 139, 1–8.
Dimentberg, M. F. (1971). “Oscillations of a system with nonlinear cubic characteristics under narrow band random excitation.” Mech. Solids, 6, 142–146.
Gillespie, D. T. (1992). Markov processes, Academic, San Diego.
Gottlieb, O., and Yim, S. C. (1992). “Nonlinear oscillations, bifurcations and chaos in a multipoint mooring system with a geometric nonlinearity.” Appl. Ocean. Res., 24, 1–257.
Guckenheimer, J., and Holmes, P. (1986). Nonlinear oscillations, dynamical systems, and bifurcation of vector fields, Springer, New York.
Jordan, D. W., and Smith, P. (1999). Nonlinear ordinary differential equations, 3rd Ed., Clarendon, Oxford, U.K.
Koliopulos, P. K., and Bishop, S. R. (1993). “Quasi-harmonic analysis of the behavior of a hardening dulling oscillator subjected to filtered white noise.” Nonlinear Dyn., 4, 279–288.
Langley, R. S. (1986). “On various definitions of the envelope of a random process.” J. Sound Vib., 105, 503–512.
Lin, H., and Yim, S. C. S. (1997). “Noisy nonlinear motions of moored systems. Part I: Analysis and simulation.” J. Waterway, Port, Coastal, Ocean Eng., 123(5), 287–295.
Lin, Y. K. (1967). Probabilistic theory of structural dynamics, McGraw-Hill, New York.
Lutes, L. D., and Sarkani, S. (1997). Stochastic analysis of structural and mechanical vibrations, Prentice-Hall, Upper Saddle River, N.J.
Lyon, R. H., Heckl, M., and Hazelgrove, C. B. (1961). “Response of hard-spring oscillator to narrow-band excitation.” J. Acoust. Soc. Am., 33, 1401–1411.
Naess, A. (1994). “Prediction of extreme response of nonlinear oscillators subjected to random loading using the path integral solution technique.” J. Res. Natl. Inst. Stand. Technol., 99(4), 465–474.
Naess, A. (1997). “A state-of-the-art report on stochastic dynamics-Section 3.2 path integral methods.” Probab. Eng. Mech., 12(4), 197–321.
Naess, A., and Johnsen, J. M. (1993). “Response statistics of nonlinear, compliant offshore structures by the path integral solution method.” Probab. Eng. Mech., 8(2), 91–106.
Naess, A., and Moe, V. (2000). “Efficient path integral methods for nonlinear dynamic systems.” Probab. Eng. Mech., 15(2), 221–231.
Nayfeh, A. H., and Mook, D. T. (1979). Nonlinear oscillations, Wiley, New York.
Newland, D. E. (1993). An introduction to random vibrations: Spectral and wavelet analysis, 3rd Ed., Longman, New York.
Nigam, N. C. (1983). Introduction to random vibrations, MIT Press, London.
Ochi, M. K. (1990). Applied probability and stochastic processes in engineering and physical sciences, Wiley, New York.
Rice, S. O. (1954). Selected papers on noise and stochastic processes, Mathematical analysis of random noise, Dover, New York.
Richard, K., and Anand, G. V. (1983). “Multiple time scaling of the response of a duffing oscillator to narrow-band random excitation.” J. Sound Vib., 86, 85–98.
Roberts, J. B., and Spanos, P. D. (1986). “Stochastic averaging: An approximate method of solving random vibration problems.” Int. J. Non-Linear Mech., 21, 111–134.
Roberts, J. B., and Spanos, P. D. (1990). Random vibration and statistical linearization, Wiley, New York.
Soong, T. T., and Grigoriu, M. (1993). Random vibration of mechanical and structural systems, Prentice-Hall, Englewood Cliffs, N.J.
Stratonovich, R. L. (1963). Topics in the theory of random noise, Gordon and Breach, New York.
Thompson, J. M. T., and Stewart, H. B. (1986). Nonlinear dynamics and chaos, Wiley, New York.
Yim, S. C. S., and Lin, H. (1992). “Probabilistic analysis of a chaotic dynamical system.” Applied chaos, Wiley, New York, 219–241.
Yim, S. C. S., Nakhata, T., Bartel, W. A., and Huang, E. T. (2005a). “Coupled nonlinear barge motions, Part I: Deterministic models development, identification and calibration.” J. Offshore Mech. Arct. Eng., 127, 1–10.
Yim, S. C. S., Nakhata, T., and Huang, E. T. (2005b). “Coupled nonlinear barge motions, Part II: Stochastic models and stability analysis.” J. Offshore Mech. Arct. Eng., 127, 83–95.
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© 2007 ASCE.
History
Received: Dec 19, 2005
Accepted: Apr 19, 2006
Published online: Jan 1, 2007
Published in print: Jan 2007
Notes
Note. Associate Editor: Ross Barry Corotis
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