Modeling Nonlinear Systems by Volterra Series
Publication: Journal of Engineering Mechanics
Volume 136, Issue 6
Abstract
The Volterra-series expansion is widely employed to represent the input-output relationship of nonlinear dynamical systems. This representation is based on the Volterra frequency-response functions (VFRFs), which can either be estimated from observed data or through a nonlinear governing equation, when the Volterra series is used to approximate an analytical model. In the latter case, the VFRFs are usually evaluated by the so-called harmonic probing method. This operation is quite straightforward for simple systems but may reach a level of such complexity, especially when dealing with high-order nonlinear systems or calculating high-order VFRFs, that it may loose its attractiveness. An alternative technique for the evaluation of VFRFs is presented here with the goal of simplifying and possibly automating the evaluation process. This scheme is based on first representing the given system by an assemblage of simple operators for which VFRFs are readily available, and subsequently constructing VFRFs of the target composite system by using appropriate assemblage rules. Examples of wind and wave-excited structures are employed to demonstrate the effectiveness of the proposed technique.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The funding for this work was provided in part by a grant from NSF (Grant No. CMMI0928282).
References
Amblard, P. O., Gaeta, M., and Lacoume, J. L. (1996). “Statistics for complex random variables and signals. Part I: Variables.” Signal Process., 53, 1–13.
Bedrosian, E., and Rice, S. O. (1971). “The output of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs.” Proc. IEEE, 59(12), 1688–1707.
Carassale, L., and Karrem, A. (2003). “Dynamic analysis of complex nonlinear systems by Volterra approach.” Proc., Computational Stochastic Mechanics, CSM4, P. D. Spanos and G. Deodatis, eds., Millipress, Rotterdam, The Netherlands, 107–117.
Carassale, L., and Solari, G. (2006). “Monte Carlo simulation of wind velocity fields on complex structures.” J. Wind Eng. Ind. Aerodyn., 94, 323–339.
Chakrabarti, S. K. (1990). Nonlinear methods in offshore engineering, Elsevier, Amsterdam.
Choi, D., Miksad, R. W., and Powers, E. J. (1985). “Application of digital cross-bispectral analysis techniques to model the non-linear response of a moored vessel system in random seas.” J. Sound Vib., 99(3), 309–326.
Donley, M. G., and Spanos, P. D. (1990). Dymanic analysis of non-linear structures by the method of statistical quadratization, Springer, New York.
Donley, M. G., and Spanos, P. D. (1991). “Stochastic response of a tension leg platform to viscous drift forces.” J. Offshore Mech. Arct. Eng., 113(2), 148–155.
Grigoriu, M. (1984). “Crossings of non-Gaussian translation processes.” J. Eng. Mech., 110(4), 610–620.
Kareem, A., and Li, Y. (1994). “Stochastic response of a tension leg platform to viscous and potential drift forces.” Probab. Eng. Mech., 9(1–2), 1–14.
Kareem, A., Williams, A. N., and Hsieh, C. C. (1994). “Diffraction of nonlinear random waves by a vertical cylinder in deep water.” Ocean Eng., 21(2), 129–154.
Kareem, A., Zhao, J., and Tognarelli, M. A. (1995). “Surge response statistics of tension leg platforms under wind & wave loads: a statistical quadratization approach.” Probab. Eng. Mech., 10(4), 225–240.
Koukoulas, P., and Kalouptsidis, N. (1995). “Nonlinear system identification using Gaussian inputs.” IEEE Trans. Signal Process., 43, 1831–1841.
Li, X. M., Quek, S. T., and Koh, C. G. (1995). “Stochastic response of offshore platforms by statistical cubicization.” J. Eng. Mech., 121(10), 1056–1068.
Li, Y., and Kareem, A. (1993). “Multivariate Hermite expansion of hydrodynamic drag loads on tension leg platforms.” J. Eng. Mech., 119(1), 91–112.
Lucia, D. J., Beranb, P. S., and Silva, W. A. (2004). “Reduced-order modeling: New approaches for computational physics.” Prog. Aerosp. Sci., 40, 51–117.
Peyton Jones, J. C. (2007). “Simplified computation of the Volterra frequency response functions of non-linear systems.” Mech. Syst. Signal Process., 21, 1452–1468.
Priestley, M. B. (1981). Spectral analysis and time series, Academic, San Diego.
Schetzen, M. (1980). The Volterra and Weiner theories of nonlinear systems, Wiley, New York.
Silva, W. (2005). “Identification of nonlinear aeroelastic systems based on the Volterra theory: progress and opportunities.” Nonlinear Dyn., 39, 25–62.
Solari, G., and Piccardo, G. (2001). “Probabilistic 3-D turbulence modeling for gust buffeting of structures.” Probab. Eng. Mech., 16, 73–86.
Spanos, P. D. and Donley, M. G. (1991). “Equivalent statistical quadratization for nonlinear systems.” J. Eng. Mech., 117(6), 1289–1310.
Spanos, P. D., and Donley, M. G. (1992). “Non-linear multi-degree-of-freedom system random vibration by equivalent statistical quadratization.” Int. J. Non-Linear Mech., 27(5), 735–748.
Spanos, P. D., Failla, G., and Di Paola, M. (2003). “Spectral approach to equivalent statistical quadratization and cubicization methods for nonlinear oscillators.” J. Eng Mech., 129(1), 31–42.
Tognarelli, M., and Kareem, A. (2001). “Equivalent cubicization for nonlinear systems.” Proc. 8th Int. Conf. on Structural Safety and Reliability, ICOSSAR (CD-ROM), Lisse, Balkema, The Netherlands.
Tognarelli, M. A., Zhao, J., and Kareem, A. (1997b). “Equivalent statistical cubicization for force and system nonlinearities.” J. Eng. Mech., 123(8), 890–893.
Tognarelli, M. A., Zhao, J., Rao, K. < middlename>B., and Kareem, A. (1997a). “Equivalent statistical quadratization and cubicization for nonlinear systems.” J. Eng Mech., 123(5), 512–523.
Volterra, V. (2005). “Theory of functionals and of integral and integrodifferential equations.” Dover, New York.
Winterstein, S. R. (1985). “Non-normal response and fatigue damage.” J. Eng. Mech., 111(10), 1291–1295.
Winterstein, S. R. (1988). “Nonlinear vibration models for extremes and fatigue.” J. Eng. Mech., 114(10), 1772–1790.
Worden, K., and Manson, G. (2005). “A Volterra series approximation to the coherence of the Duffing oscillator.” J. Sound Vib., 286, 529–547.
Worden, K., Manson, G., and Tomlinson, G. R. (1997). “A harmonic probing algorithm for the multi-input Volterra series.” J. Sound Vib., 201(1), 67–84.
Young, I. R. (1999). Wind generated ocean waves, Elsevier, Oxford.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 136 • Issue 6 • June 2010
Pages: 801 - 818
Copyright
© 2010 ASCE.
History
Received: May 18, 2009
Accepted: Nov 5, 2009
Published online: Nov 6, 2009
Published in print: Jun 2010
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.