Abstract
The nonstationary probability densities of system response of a single-degree-of-freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied. Using the stochastic averaging method based on the generalized harmonic functions, the averaged Fokker-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude is derived. The solution of the equation is approximated by the series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. According to the Galerkin method, the time-dependent coefficients can be solved from a set of first-order linear differential equations. Then, the semi-analytical formulae of the nonstationary probability density of the amplitude response as well as the nonstationary probability density of the state response and the statistic moments of the amplitude response can be obtained. A van der Pol-Duffing oscillator subject to modulated white noise is given as an example to illustrate the proposed procedures. The effects of the system parameters, such as the linear damping coefficient and the nonlinear stiffness coefficient, on the system response are discussed.
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Wirsching, P. H. and Yao, J. T. P. Monte Carlo study of seismic structural safety. ASCE Journal of the Structural Division, 97, 1497–1519 (1971)
Howell, L. J. and Lin, Y. K. Response of flight vehicles to nonstationary atmospheric turbulence. AIAA Journal, 9, 2201–2207 (1971)
Yang, Y. N. Nonstationary envelope process and first excursion probability. Journal of Structural Mechanics, 1, 231–248 (1972)
Fujimori, Y. and Lin, Y. K. Analysis of airplane response to nonstationary turbulence including wing bending flexibility. AIAA Journal, 11, 334–339 (1973)
Zhang, Z. C., Lin, J. H., Zhang, Y. H., Zhao, Y., Howson, W. P., and Williams, F. W. Nonstationary random vibration analysis for train-bridge systems subjected to horizontal earthquakes. Engineering Structures, 32, 3571–3582 (2010)
Abbas, A. M. and Manohar, C. S. Reliability-based vector nonstationary random critical earthquake excitations for parametrically excited systems. Structural Safety, 29, 32–48 (2007)
Caughey, T. K. and Stumpf, H. F. Transient response of a dynamic system under random excitation. Journal of Applied Mechanics-Transactions of the ASME, 28, 563–566 (1961)
Corotis, R. B. and Marshall, T. A. Oscillator response to modulated random-excitation. Journal of the Engineering Mechanics Division-ASCE, 103, 501–513 (1977)
Iwan, W. D. and Hou, Z. K. Explicit solutions for the response of simple systems subjected to nonstationary random-excitation. Structural Safety, 6, 77–86 (1989)
Fu, G. Seismic response statistics of SDOF system to exponentially modulated coloured input: an explicit solution. Earthquake Engineering & Structural Dynamics, 24, 1355–1370 (1995)
Michaelov, G., Sarkani, S., and Lutes, L. D. Spectral characteristics of nonstationary random processes response of a simple oscillator. Structural Safety, 21, 245–267 (1999)
Jangid, R. S. Stochastic response of building frames isolated by lead-rubber bearings. Structural Control & Health Monitoring, 17, 1–22 (2010)
Allam, S. M. and Datta, T. K. Seismic response of a cable-stayed bridge deck under multicomponent non-stationary random ground motion. Earthquake Engineering & Structural Dynamics, 33, 375–393 (2004)
Verdon, J. M. Response of a single-degree-of-freedom system to modulated white noise. Journal of Applied Mechanics-Transactions of the ASME, 40, 296–297 (1973)
To, C. W. S. Non-stationary random responses of a multi-degree-of-freedom system by the theory of evolutionary spectra. Journal of Sound and Vibration, 83, 273–291 (1982)
Ahmadi, G. Mean square response of a Duffing oscillator to a modulated white noise excitation by the generalized method of equivalent linearization. Journal of Sound and Vibration, 71, 9–15 (1980)
Iwan, W. D. and Mason, A. B. Equivalent linearization for systems subjected to non-stationary random excitation. International Journal of Non-Linear Mechanics, 15, 71–82 (1980)
Fang, T. and Zhang, T. S. Non-stationary mean-square response due to uniformly amplitude modulated random excitations. Journal of Sound and Vibration, 182, 369–379 (1995)
Spanos, P. D. A method for analysis of non-linear vibrations caused by modulated random-excitation. International Journal of Non-Linear Mechanics, 16, 1–11 (1981)
Kougioumtzoglou, I. A. and Spanos, P. D. An approximate approach for nonlinear system response determination under evolutionary stochastic excitation. Current Science, 97, 1203–1211 (2009)
Zhu, W. Q. Nonlinear stochastic dynamics and control in Hamiltonian formulation. Applied Mechanics Reviews, 59, 230–248 (2006)
Zhu, W. Q., Huang, Z. L., and Suzuki, Y. Response and stability of strongly non-linear oscillators under wide-band random excitation. International Journal of Non-Linear Mechanics, 36, 1235–1250 (2001)
Xu, Z. and Chung, Y. K. Averaging method using generalized harmonic functions for strongly non-linear oscillators. Journal of Sound and Vibration, 174, 563–576 (1994)
Khasminskii, R. Z. On the averaging principle for Itô stochastic differential equations (in Russian). Kibernetika, 4, 260–279 (1968)
Spanos, P. D. and Iwan, W. D. Computational aspects of random vibration analysis. Journal of the Engineering Mechanics Division-ASCE, 104, 1403–1415 (1978)
Spanos, P. D., Sofi, A., and di Paola, M. Nonstationary response envelope probability densities of nonlinear oscillators. Journal of Applied Mechanics-Transactions of the ASME, 74, 315–324 (2007)
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Project supported by the National Natural Science Foundation of China (No. 11025211), the Zhejiang Provincial Natural Science Foundation of China (No. Z6090125), and the Special Fund for National Excellent Ph. D. Dissertation and Research Grant Council of Hong Kong City (No.U115807)
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Jin, Xl., Huang, Zl. & Leung, Y.T. Nonstationary probability densities of system response of strongly nonlinear single-degree-of-freedom system subject to modulated white noise excitation. Appl. Math. Mech.-Engl. Ed. 32, 1389–1398 (2011). https://doi.org/10.1007/s10483-011-1509-7
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DOI: https://doi.org/10.1007/s10483-011-1509-7