Time-Domain Response of Damped Stochastic Multiple-Degree-of-Freedom Systems
Publication: Journal of Engineering Mechanics
Volume 146, Issue 1
Abstract
Characterizing the time-domain response of a random multiple-degree-of-freedom dynamical system is challenging and often requires Monte Carlo simulation (MCS). Differential equations must therefore be solved for each sample, which is time-consuming. This is why polynomial chaos expansion (PCE) has been proposed as an alternative to MCS. However, it turns out that PCE is not adapted to simulate a random dynamical system for long-time integration. Recent studies have shown similar issues for the frequency response function of a random linear system around the deterministic eigenfrequencies. A Padé approximant approach has been successfully applied; similar interesting results were also observed with a random mode approach. Therefore, the latter two methods were applied to a random linear dynamical system excited by a dynamic load to estimate the first two statistical moments and probability density function at a given instant of time. Whereas the random modes method has been very efficient and accurate to evaluate the statistics of the response, the Padé approximant approach has given very poor results when the coefficients were determined in the time domain. However, if the differential equations were solved in the frequency domain, the Padé approximants, which were also calculated in the frequency domain, provided results in excellent agreement with the MCS results.
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References
Adhikari, S. 2011. “A reduced spectral function approach for the stochastic finite element analysis.” Comput. Methods Appl. Mech. Eng. 200 (21–22): 1804–1821. https://doi.org/10.1016/j.cma.2011.01.015.
Baker, G. A., and P. Graves-Morris. 1996. Padé approximants. 2nd ed. Cambridge, UK: Cambridge University Press.
Basseville, M. 2013. “Divergence measures for statistical data processing: An annotated bibliography.” Signal Process. 93 (4): 621–633. https://doi.org/10.1016/j.sigpro.2012.09.003.
Beran, P., C. Pettit, and D. Millman. 2006. “Uncertainty quantification of limit-cycle oscillations.” J. Comput. Phys. 217 (1): 217–247. https://doi.org/10.1016/j.jcp.2006.03.038.
Chantrasmi, T., A. Doostan, and G. Iaccarino. 2009. “Padé–Legendre approximants for uncertainty analysis with discontinuous response surfaces.” J. Comput. Phys. 228 (19): 7159–7180. https://doi.org/10.1016/j.jcp.2009.06.024.
Chatterjee, T., S. Chakraborty, and R. Chowdhury. 2016. “A bi-level approximation tool for the computation of FRFs in stochastic dynamic systems.” Mech. Syst. Signal Process. 70–71 (Mar): 484–505. https://doi.org/10.1016/j.ymssp.2015.09.001.
Dessombz, O., A. Diniz, F. Thouverez, and L. Jézéquel. 1999. “Analysis of stochastic structures: Pertubation method and projection on homogeneous chaos.” In Proc., 7th Int. Modal Analysis Conf. IMAC-SEM. Bethel, CT: Society for Experimental Mechanics.
Doyle, J. F. 1997. Wave propagation in structures: Spectral analysis using fast discrete Fourier transforms. 2nd ed. Berlin: Springer.
Emmel, L., S. M. Kaber, and Y. Maday. 2003. “Padé-Jacobi filtering for spectral approximations of discontinuous solutions.” Numer. Algorithms 33 (1): 251–264. https://doi.org/10.1023/A:1025572207222.
Fricker, T. E., J. E. Oakley, N. D. Sims, and K. Worden. 2011. “Probabilistic uncertainty analysis of an FRF of a structure using a Gaussian process emulator.” Mech. Syst. Sig. Process. 25 (8): 2962–2975. https://doi.org/10.1016/j.ymssp.2011.06.013.
Gerritsma, M., J.-B. van der Steen, P. Vos, and G. Karniadakis. 2010. “Time-dependent generalized polynomial chaos.” J. Comput. Phys. 229 (22): 8333–8363. https://doi.org/10.1016/j.jcp.2010.07.020.
Ghanem, R. G., and P. D. Spanos. 1991. Stochastic finite elements: A spectral approach. New York: Springer.
Goller, B., H. Pradlwarter, and G. Schueller. 2011. “An interpolation scheme for the approximation of dynamical systems.” Comput. Methods Appl. Mech. Eng. 200 (1–4): 414–423. https://doi.org/10.1016/j.cma.2010.09.005.
Guillaume, P., A. Huard, and V. Robin. 2000. “Multivariate Padé approximation.” J. Comput. Appl. Math. 121 (1–2): 197–219. https://doi.org/10.1016/S0377-0427(00)00337-X.
Jacquelin, E., S. Adhikari, J.-J. Sinou, and M. I. Friswell. 2015. “The polynomial chaos expansion and the steady-state response of a class of random dynamic systems.” J. Eng. Mech. 141 (4): 04014145. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000856.
Jacquelin, E., O. Dessombz, J.-J. Sinou, S. Adhikari, and M. I. Friswell. 2017. “Polynomial chaos based extended Padé expansion in structural dynamics.” Int. J. Numer. Methods Eng. 111 (12): 1170–1191. https://doi.org/10.1002/nme.5497.
Kullback, S., and R. A. Leibler. 1951. “On information and sufficiency.” Ann. Math. Stat. 22 (1): 79–86. https://doi.org/10.1214/aoms/1177729694.
Le Maître, O., L. Mathelin, O. Knio, and M. Hussaini. 2010. “Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics.” Discret. Contin. Dyn. Sys. 28 (1): 199–226. https://doi.org/10.3934/dcds.2010.28.199.
Le Meitour, J., D. Lucor, and J.-C. Chassaing. 2010. “Prediction of stochastic limit cycle oscillations using an adaptive polynomial chaos method.” J. Aeroelasticity Struct. Dyn. 2 (1): 3–22. https://doi.org/10.3293/asdj.2010.4.
Mai, C., M. Spiridonakos, E. Chatzi, and B. Sudret. 2016. “Surrogate modeling for stochastic dynamical systems by combining nonlinear autoregressive with exogenous input models and polynomial chaos expansions.” Int. J. Uncertainty Quantif. 6 (4): 313–339. https://doi.org/10.1615/Int.J.UncertaintyQuantification.2016016603.
Mai, C., and B. Sudret. 2017. “Surrogate models for oscillatory systems using sparse polynomial chaos expansions and stochastic time warping.” J. Uncertainty Quantif. 5 (1): 540–571. https://doi.org/10.1137/16M1083621.
Matos, A. C. 1996. “Some convergence results for the generalized Padé-type approximants.” Numer. Algorithms 11 (1): 255–269. https://doi.org/10.1007/BF02142501.
Sarrouy, E., E. Pagnacco, and E. S. de Cursi. 2016. “A constant phase approach for the frequency response of stochastic linear oscillators.” Mech. Ind. 17 (2): 206–215. https://doi.org/10.1051/meca/2015057.
Siqueira Meirelles, P., and J. Arruda. 2005. “Transient response with arbitrary initial conditions using the DFT.” J. Sound Vib. 287 (3): 525–543. https://doi.org/10.1016/j.jsv.2004.11.007.
Wiener, N. 1938. “The homogeneous chaos.” Am. J. Math. 60 (4): 897–936. https://doi.org/10.2307/2371268.
Witteveen, J., and H. Bijl. 2006. “Modeling arbitrary uncertainties using Gram-Schmidt polynomial chaos.” In Proc., 44th AIAA Aerospace Sciences Meeting. Reston, VA: American Institute of Aeronautics and Astronautics.
Xiu, D. 2010. Numerical methods for stochastic computations: A spectral method approach. Princeton, NJ: Princeton University Press.
Xiu, D., and G. Karniadakis. 2002. “The Wiener-Askey polynomial chaos for stochastic differential equations.” SIAM J. Sci. Comput. 24 (2): 619–644. https://doi.org/10.1137/S1064827501387826.
Yaghoubi, V., S. Marelli, B. Sudret, and T. Abrahamsson. 2017. “Sparse polynomial chaos expansions of frequency response functions using stochastic frequency transformation.” Probab. Eng. Mech. 48 (Apr): 39–58. https://doi.org/10.1016/j.probengmech.2017.04.003.
Yamazaki, F., M. Shinozuka, and G. Dasgupta. 1988. “Neumann expansion for stochastic finite element analysis.” J. Eng. Mech. 114 (8): 1335–1354. https://doi.org/10.1061/(ASCE)0733-9399(1988)114:8(1335).
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Published In
Journal of Engineering Mechanics
Volume 146 • Issue 1 • January 2020
Copyright
©2019 American Society of Civil Engineers.
History
Received: Aug 21, 2018
Accepted: Jun 5, 2019
Published online: Nov 14, 2019
Published in print: Jan 1, 2020
Discussion open until: Apr 14, 2020
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