Responses of Nonlinear Oscillators Excited by Nonzero-Mean Parametric Poisson Impulses on Displacement
Publication: Journal of Engineering Mechanics
Volume 138, Issue 5
Abstract
The nonzero-mean probability density function (PDF) solutions of nonlinear stochastic oscillators under the excitation of Poisson impulse process are obtained with exponential-polynomial closure (EPC) method. The excitations are assumed to be external Poisson impulse process and parametric Poisson impulse process on displacement. The PDF of the oscillator response is governed by the generalized Fokker-Planck-Kolmogorov (FPK) equation, which is solved with the EPC method. The nonlinear oscillator with external and parametric excitation on displacement is analyzed when the mean of oscillator response is nonzero. Different levels of oscillator nonlinearity and nonzero means of the impulse amplitude are considered in the analysis. The analytical results show that the PDF solutions given by the EPC method are in good agreement with the simulated results when the complete sixth-degree polynomial of state variables is taken in the EPC procedure. The good agreement is also observed in the tail regions of the PDF solutions. The numerical analysis further shows that the PDF of displacement is not symmetrical about the mean of displacement.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The authors are grateful to the valuable suggestions and discussions from Professor R. Iwankiewicz that make this paper more publishable. This research is supported by a grant from the National Natural Science Foundation of China (Grant No. 51008211), a grant from the Self-Innovation Foundation of Tianjin University (Grant No. 60302024), and a grant from the Research Committee of the University of Macau (Grant No. MYRG138(Y1-L2)-FST11-EGK).
References
Bergman, L. A., Spencer, B. F. Jr., Wojtkiewicz, S. F., and Johnson, E. A. (1996). “Robust numerical solution of the Fokker-Planck equation for second order dynamical systems under parametric and external white noise excitations.” Nonlinear dynamics and stochastic mechanics, Kliemann, W. H., Langford, W. F., and Sri Namachchivaya, N., eds., American Mathematical Society, Providence, RI, 23–37.
Cai, G. Q., and Lin, Y. K. (1992). “Response distribution of non-linear systems excited by non-Gaussian impulsive noise.” Int. J. Nonlinear Mech.IJNMAG, 27(6), 955–967.
Di Paola, M., and Falsone, G. (1994). “Non-linear oscillators under parametric and external Poisson pulses.” Nonlinear Dyn.NODYES, 5(3), 337–352.
Elishakoff, I., Fang, J., and Caimi, R. (1995). “Random vibration of a nonlinearly deformed beam by a new stochastic linearization technique.” Int. J. Solids Struct.IJSOAD, 32(11), 1571–1584.
Er, G. K. (1998). “A new non-Gaussian closure method for the PDF solution of nonlinear random vibrations.” Proc., 12th Engineering Mechanics Div. Conf., ASCE, Reston, VA, 1403–1406.
Er, G. K. (1999). “Consistent method for PDF solutions of random oscillators.” J. Eng. Mech.JENMDT, 125(4), 443–447.
Er, G. K.(2011). “Methodology for the solutions of some reduced Fokker-Planck equations in high dimensions.” Ann. Phys.ANPYA2, 523(3), 247–258.
Er, G. K., and Iu, V. P. (1999). “Probabilistic solutions to nonlinear random ship roll motion.” J. Eng. Mech.JENMDT, 125(5), 570–574.
Er, G. K., and Iu, V. P. (2011). “A new method for the probabilistic solutions of large-scale nonlinear stochastic dynamic systems.” IUTAM Symp. on Nonlinear Stochastic Dynamics and Control, IUTAM Bookseries, Zhu, W. Q., Lin, Y. K., and Cai, G. Q., eds., Vol. 29, Springer, New York, 25–34.
Grigoriu, M. (1987). “White noise processes.” J. Eng. Mech.JENMDT, 113(5), 757–765.
Grigoriu, M. (1995). “Equivalent linearization for Poisson white noise input.” Prob. Eng. Mech.PEMEEX, 10(1), 45–51.
Hu, S. L. J. (1993). “Responses of dynamic systems excited by non-Gaussian pulse processes.” J. Eng. Mech.JENMDT, 119(9), 1818–1827.
Hu, S. L. J. (1995). “Parametric random vibrations under non-Gaussian delta-correlated processes.” J. Eng. Mech.JENMDT, 121(12), 1366–1371.
Iwankiewicz, R., and Nielsen, S. R. K. (1992a). “Dynamic response of non-linear systems to Poisson-distributed random impulses.” J. Sound Vib.JSVIAG, 156(3), 407–423.
Iwankiewicz, R., and Nielsen, S. R. K. (1992b). “Dynamic response of hysteretic systems to Poisson-distributed pulse trains.” Prob. Eng. Mech.PEMEEX, 7(3), 135–148.
Iwankiewicz, R., Nielsen, S. R. K., and Thoft-Christensen, P. (1990). “Dynamic response of non-linear systems to Poisson-distributed pulse trains: Markov approach.” Struct. Saf., 8(1–4), 223–238.
Kovacic, I., Brennan, M. J. (2011). The Duffing equation: Nonlinear oscillators and their behaviour, Wiley, West Sussex, United Kingdom.
Köylüogˇlu, H. U., Nielsen, S. R. K., and Iwankiewicz, R. (1994). “Reliability of non-linear oscillators subject to Poisson driven impulses.” J. Sound Vib.JSVIAG, 176(1), 19–33.
Köylüogˇlu, H. U., Nielsen, S. R. K., and Iwankiewicz, R. (1995a). “Response and reliability of Poisson-driven systems by path integration.” J. Eng. Mech.JENMDT, 121(1), 117–130.
Köylüogˇlu, H. U., Nielsen, S. R. K., and Çakmak, A. Ş. (1995b). “Fast cell-to-cell mapping (path integration) for nonlinear white noise and Poisson driven systems.” Struct. Saf., 17(3), 151–165.
Lin, Y. K. (1986). “Some observations on the stochastic averaging method.” Prob. Eng. Mech.PEMEEX, 1(1), 23–27.
Proppe, C. (2002). “Equivalent linearization of MDOF systems under external Poisson white noise excitation.” Prob. Eng. Mech.PEMEEX, 17(4), 393–399.
Proppe, C. (2003). “Exact stationary probability density functions for non-linear systems under Poisson white noise excitation.” Int. J. Nonlinear Mech.IJNMAG, 38(4), 557–564.
Roberts, J. B. (1972). “System response to random impulses.” J. Sound Vib.JSVIAG, 24(1), 23–34.
Roberts, J. B., and Spanos, P. D. (2003). Random vibration and statistical linearization, Dover Publications, Mineola, NY.
Tylikowski, A., and Marowski, W. (1986). “Vibration of a non-linear single degree of freedom system due to Poissonian impulse excitation.” Int. J. Non-linear Mech.IJNMAG, 21(3), 229–238.
Vasta, M. (1995). “Exact stationary solution for a class of nonlinear systems driven by a non-normal delta-correlated process.” Int. J. Non-linear Mech.IJNMAG, 30(4), 407–418.
Vasta, M., and Luongo, A. (2004). “Dynamic analysis of linear and nonlinear oscillations of a beam under axial and transversal random Poisson pulses.” Nonlinear Dyn.NODYES, 36(2–4), 421–435.
Wojtkiewicz, S. F., Johnson, E. A., Bergman, L. A., Grigoriu, M., and Spencer, B. F. Jr. (1999). “Response of stochastic dynamical systems driven by additive Gaussian and Poisson white noise: Solution of a forward generalized Kolmogorov equation by a spectral finite difference method.” Comput. Methods Appl. Mech. Eng., 168(1–4), 73–89.
Zeng, Y., and Zhu, W. Q. (2010). “Stochastic averaging of quasi-linear systems driven by Poisson white noise.” Prob. Eng. Mech.PEMEEX, 25(1), 99–107.
Zhu, H. T., Er, G. K., Iu, V. P., and Kou, K. P. (2010). “Probability density function solution of nonlinear oscillators subjected to multiplicative Poisson pulse excitation on velocity.” J. Appl. Mech.JAMCAV, 77(3), 031001.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 138 • Issue 5 • May 2012
Pages: 450 - 457
Copyright
© 2012. American Society of Civil Engineers.
History
Received: Sep 8, 2010
Accepted: Nov 16, 2011
Published online: Nov 18, 2011
Published in print: May 1, 2012
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.