We study first-crossing problem of two-degrees-of-freedom (2DOF) strongly nonlinear mechanical oscillators analytically. The excitation is the combination of a deterministic harmonic function and Gaussian white noises (GWNs). The generalized harmonic function is used to approximate the solutions of the original equations. Four cases are studied in terms of the types of resonance (internal or external or both). For each case, the method of stochastic averaging is used and the stochastically averaged Itô equations are obtained. A backward Kolmogorov (BK) equation is set up to yield the failure probability and a Pontryagin equation is set up to yield average first-crossing time (AFCT). A 2DOF Duffing-van der Pol oscillator is chosen as an illustrative example to demonstrate the effectiveness of the analytical method. Numerically analytical solutions are obtained and validated by digital simulation. It is shown that the proposed method has high efficiency while still maintaining satisfactory accuracy.
Issue Section:
Research Papers
References
1.
Hamed
, Y. S.
, EL-Sayed
, A. T.
, and El-Zahar
, E. R.
, 2016
, “On Controlling the Vibrations and Energy Transfer in MEMS Gyroscope System With Simultaneous Resonance
,” Nonlinear Dyn.
, 83
(3
), pp. 1687
–1704
.2.
Shoeybi
, M.
, and Ghorashi
, M.
, 2006
, “Nonlinear Vibration Control of a System With Dry Friction and Viscous Damping Using the Saturation Phenomenon
,” Nonlinear Dyn.
, 45
(3–4
), pp. 249
–272
.3.
El-Bassiouny
, A. F.
, 2005
, “Internal Resonance of a Nonlinear Vibration Absorber
,” Phys. Scr.
, 72
(2–3
), pp. 203
–211
.4.
Wang
, Y. R.
, and Liang
, T. W.
, 2015
, “Application of Lumped-Mass Vibration Absorber on the Vibration Reduction of a Nonlinear Beam-Spring-Mass System With Internal Resonances
,” J. Sound Vib.
, 350
, pp. 140
–170
.5.
Ji
, J. C.
, 2014
, “Design of a Nonlinear Vibration Absorber Using Three-to-One Internal Resonances
,” Mech. Syst. Signal Process.
, 42
(1–2
), pp. 236
–246
.6.
Song
, Y.
, Liu
, Z. G.
, Duan
, F. C.
, Lu
, X. B.
, and Wang
, H. R.
, 2017
, “Study on Wind-Induced Vibration Behavior of Railway Catenary in Spatial Stochastic Wind Field Based on Nonlinear Finite Element Procedure
,” ASME J. Vib. Acoust.
, 140
(1
), p. 011010
.7.
Cox
, D. R.
, and Miller
, H. D.
, 1965
, The Theory of Stochastic Processes
, Chapman and Hall
, New York
.8.
Huang
, Z. L.
, and Zhu
, W. Q.
, 1997
, “Exact Stationary Solutions of Averaged Equations of Stochastically and Harmonically Excited MDOF Quasi-Linear Systems With Internal and or External Resonances
,” J. Sound Vib.
, 204
(2
), pp. 249
–258
.9.
Huang
, Z. L.
, and Zhu
, W. Q.
, 2004
, “Stochastic Averaging of Quasi-Integrable Hamiltonian Systems Under Combined Harmonic and White Noise Excitations
,” Int. J. Non-Linear Mech.
, 39
(9
), pp. 1421
–1434
.10.
Xu
, W.
, He
, Q.
, Rong
, H. W.
, and Fang
, T.
, 2002
, “One-to-Two Internal Resonance in Two-Degrees-of-Freedom Nonlinear System With Narrow-Band Excitations
,” Nonlinear Dyn.
, 27
(4
), pp. 385
–395
.11.
Ikeda
, T.
, and Ibrahim
, R. A.
, 2005
, “Nonlinear Random Responses of a Structure Parametrically Coupled With Liquid Sloshing in a Cylindrical Tank
,” J. Sound Vib.
, 284
(1–2
), pp. 75
–102
.12.
Sapsis
, T. P.
, Vakakis
, A. F.
, and Bergman
, L. A.
, 2011
, “Effect of Stochasticity on Targeted Energy Transfer From a Linear Medium to a Strongly Nonlinear Attachment
,” Probab. Eng. Mech.
, 26
(2
), pp. 119
–133
.13.
Chen
, L. Q.
, and Jiang
, W. A.
, 2015
, “Internal Resonance Energy Harvesting
,” ASME J. Appl. Mech.
, 82
(3
), p. 031004
.14.
Jia
, W. T.
, and Zhu
, W. Q.
, 2014
, “Stochastic Averaging of Quasi-Integrable and Non-Resonant Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations
,” Nonlinear Dyn.
, 76
(2
), pp. 1271
–1289
.15.
Jia
, W. T.
, and Zhu
, W. Q.
, 2013
, “Stochastic Averaging of Quasi-Integrable and Resonant Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations
,” ASME J. Appl. Mech.
, 81
(4), p. 041009
.16.
Sun
, J. Q.
, and Hsu
, C. S.
, 1988
, “First-Passage Time Probability of Non-Linear Stochastic Systems by Generalized Cell Mapping Method
,” J. Sound Vib.
, 124
(2
), pp. 233
–248
.17.
Au
, S. K.
, and Beck
, J. L.
, 2003
, “Important Sampling in High Dimensions
,” Struct. Saf.
, 25
(2
), pp. 139
–163
.18.
Bucher
, C.
, 2009
, “Asymptotic Sampling for High-Dimensional Reliability Analysis
,” Probab. Eng. Mech.
, 24
(4
), pp. 504
–510
.19.
Sichani
, M. T.
, Nielsen
, S. R. K.
, and Bucher
, C.
, 2011
, “Efficient Estimation of First Passage Probability of High-Dimensional Nonlinear Systems
,” Probab. Eng. Mech.
, 26
(4
), pp. 539
–549
.20.
Naess
, A.
, and Moe
, V.
, 2000
, “Efficient Path Integration Methods for Nonlinear Dynamic Systems
,” Probab. Eng. Mech.
, 15
(2
), pp. 221
–231
.21.
Yang
, J.
, Faverjon
, B.
, Peters
, H.
, Marburg
, S.
, and Kessissoglou
, N.
, 2017
, “Deterministic and Stochastic Model Order Reduction for Vibration Analyses of Structures With Uncertainties
,” ASME J. Vib. Acoust.
, 139
(2
), p. 021007
.22.
Roberts
, J. B.
, and Spanos
, P. D.
, 1986
, “Stochastic Averaging: An Approximate Method of Solving Random Vibration Problems
,” Int. J. Non-Linear Mech.
, 21
(2
), pp. 111
–134
.23.
Zhu
, W. Q.
, 1996
, “Recent Developments and Applications of Stochastic Averaging Method in Random Vibration
,” ASME Appl. Mech. Rev.
, 49
(10S
), pp. s72
–s80
.24.
Zhu
, W. Q.
, 2006
, “Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation
,” ASME Appl. Mech. Rev.
, 59
(4
), pp. 230
–248
.25.
Huang
, Z. L.
, Zhu
, W. Q.
, and Suzuki
, Y.
, 2000
, “Stochastic Averaging of Strongly Nonlinear Oscillators Under Combined Harmonic and White-Noise Excitations
,” J. Sound Vib.
, 238
(2
), pp. 233
–256
.26.
Huang
, Z. L.
, Zhu
, W. Q.
, Ni
, Y. Q.
, and Ko
, J. M.
, 2003
, “Stochastic Averaging of Strongly Non-Linear Oscillators Under Bounded Noise Excitation
,” J. Sound Vib.
, 254
(2
), pp. 245
–267
.27.
Wu
, Y. J.
, and Zhu
, W. Q.
, 2008
, “Stochastic Averaging of Strongly Nonlinear Oscillators Under Combined Harmonic and Wide-Band Noise Excitations
,” ASME J. Vib. Acoust.
, 130
(5
), p. 051004
.28.
Chen
, L. C.
, and Zhu
, W. Q.
, 2009
, “The First Passage Failure of SDOF Strongly Nonlinear Stochastic System With Fractional Derivative Damping
,” J. Vib. Control
, 15
(8
), pp. 1247
–1266
.29.
Deng
, M. L.
, and Zhu
, W. Q.
, 2007
, “Stochastic Averaging of MDOF Quasi-Integrable Hamiltonian Systems Under Wide-Band Random Excitation
,” J. Sound Vib.
, 305
(4–5
), pp. 783
–794
.30.
Wu
, Y. J.
, Luo
, M.
, and Zhu
, W. Q.
, 2008
, “First-Passage Failure of Strongly Nonlinear Oscillators Under Combined Harmonic and Real Noise Excitations
,” Arch. Appl. Mech.
, 78
(7
), pp. 501
–515
.31.
Wu
, Y. J.
, Gao
, Y. Y.
, and Zhang
, L.
, 2013
, “First-Passage Problem of Strongly Nonlinear Stochastic Oscillators With External and Internal Resonances
,” Eur. J. Mech.-A/Solids
, 39
, pp. 60
–68
.32.
Wang
, H. Y.
, and Wu
, Y. J.
, 2016
, “First-Passage Problem of a Class of Internally Resonant Quasi-Integrable Hamiltonian System Under Wide-Band Stochastic Excitations
,” Int. J. Non-Linear Mech.
, 85
, pp. 143
–151
.33.
Zhu
, H. T.
, and Guo
, S. S.
, 2015
, “Periodic Response of a Duffing Oscillator Under Combined Harmonic and Random Excitations
,” ASME J. Vib. Acoust.
, 137
(4
), p. 041015
.34.
Cveticanin
, L.
, 2015
, “A Solution Procedure Based on the Ateb Function for a Two-Degree-of-Freedom Oscillator
,” J. Sound Vib.
, 346
, pp. 298
–213
.35.
Cheung
, Y. K.
, and Xu
, Z.
, 1995
, “Internal Resonance of Strongly Non-Linear Autonomous Vibrating Systems With Many Degrees of Freedom
,” J. Sound Vib.
, 180
(2
), pp. 229
–238
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