Uncertainty Analysis in Structural Dynamics

Abstract:

This paper discusses the main issues of Uncertainty Analysis (UA) in general and also argues and illustrates its particular relevance to structural dynamics. Brief descriptions are given of the most prevalent of the many frameworks for uncertainty representation. The three main uncertainty-related problems of relevance to structural dynamics are then discussed, namely quantification, fusion and propagation. In order to illustrate the application of ideas of UA in a realistic scenario, there then follows a case study conducted on an aerospace structure, namely the wing of a Gnat trainer aircraft. The case study considers evidence-based classifiers as an alternative to probabilistic classifiers for the problem of damage location within the context of Structural Health Monitoring. Dempster-Shafer theory is employed to construct neural network classifiers with the potential to admit ignorance, rather than misclassify.

• Periodical: Key Engineering Materials (Volume 588)
• Pages: 318-332
• DOI: https://doi.org/10.4028/www.scientific.net/KEM.588.318
• Online since: October 2013
• Authors: Keith Worden
• Keywords: Convex Models, Damage Location, Evidence Theory, Fusion, Fuzzy Logic, Interval Analysis, Possibility Theory, Probability Theory, Propagation, Quantification, Uncertainty

References

[1] B.R. Mace, K. Worden and G. Manson, "Uncertainty in structural dynamics - editorial", Journal of Sound and Vibration, 2005, 288, 423-429.

DOI: 10.1016/j.jsv.2005.07.014

[2] D. Moens, "Uncertainty in structural dynamics - editorial", Mechanical Systems and Signal Processing, 2012, 32, 1-374.

[3] E.T. Jaynes, "Probability Theory: The Logic of Science: Principles and Elementary Applications: Vol 1", Cambridge University Press, 2003.

[4] C.R. Farrar and K. Worden, "Structural Health Monitoring: A Machine Learning Perspective", John Wiley and Sons, 2012.

[5] D.V. Lindley, "Introduction to Probability and Statistics from a Bayesian Viewpoint, Part 1: Probability", Cambridge University Press, 1980.

[6] A.P. Dempster, "Upper and lower probabilities induced by a multi-valued mapping", Annals of Mathematical Statistics, 2006, 38, 325-339.

DOI: 10.1214/aoms/1177698950

[7] G. Shafer G, "A Mathematical Theory of Evidence", Princeton University Press, 1976.

[8] S.C.A. Thomopoulos, "Theories in distributed decision fusion: comparison and generalisation", In: Sensor Fusion III: 3D Perception and Recognition. Proceedings of SPIE, 1383, 1990.

DOI: 10.1117/12.25302

[9] W.L. Oberkampf, J.C. Helton and K. Sentz, "Mathematical representation of uncertainty", AIAA Non-Deterministic Approaches Forum, Seattle, WA, 2001, AIAA, 2001-1645.

DOI: 10.2514/6.2001-1645

[10] R.A. Osegueda, H. Lopez, L. Pereyra and C.M. Ferregut, "Localisation of damage using fusion of modal strain energy differences", Proceedings of 18th International Modal Analysis Conference, IMAC XVIII, San Antonio, 2000, 695-701.

[11] L. Pereyra, R.A. Osegueda, C. Carrasco and C.M. Ferregut, "Detection of damage in a stiffened plate from fusion of modal strain energy differences", Proceedings of 18th International Modal Analysis Conference, IMAC XVIII, San Antonio, 2000, 1556-1562.

DOI: 10.1117/12.339887

[12] D. Dubois and H. Prade, "Possibility Theory: An Approach to Computerised Processing of Uncertainty", Plenum Press, New York, 1986.

[13] G.J. Klir, "On fuzzy-set interpretation of possibility theory", Fuzzy Sets and Systems, 1999, 108, 263-273.

DOI: 10.1016/s0165-0114(97)00371-0

[14] L.A. Zadeh, "Fuzzy sets", Information and Control, 1965, 338-353.

[15] B. Lallemand, G. Plessis, T. Tison and P. Level, "Modal behaviour of structures defined by imprecise geometric parameters", Proceedings of 18th International Modal Analysis Conference, IMAC XVIII, San Antonio, 2000, 1422-1428.

[16] G. Plessis, B. Lallemand, T. Tison and P. Level, "A fuzzy method for the modal identification of uncertain experimental data", Proceedings of 18th International Modal Analysis Conference, IMAC XVIII, San Antonio, 2000, 1831-1837.

[17] G. Plessis, B. Lallemand, T. Tison and P. Level, "Fuzzy modal parameters", Journal of Sound and Vibration, 2000, 797-812.

DOI: 10.1006/jsvi.1999.2851

[18] B. Lallemand, A. Cherki, T. Tison and P. Level, "Fuzzy modal finite element analysis of structures with imprecise material properties", Journal of Sound and Vibration, 2000, 353-364.

DOI: 10.1006/jsvi.1998.1952

[19] R.M. Moore, "Interval Analysis", Prentice Hall, 1966.

[20] J.L.D. Comba and J. Stolfi, "Affine arithmetic and its applications to computer graphics", Proceedings of SIBGRAPH '93, 1993, 9-18.

[21] G. Manson, "Sharper eigenproblem estimates for uncertain multi degree of freedom systems", Proceedings of 21st International Modal Analysis Conference, Orlando, Florida, 2003.

[22] Y. Ben-Haim and I. Elishakoff, "Convex Models of Uncertainty in Applied Mechanics", Elsevier Science Publishers, 1990.

[23] Y. Ben-Haim, "Information-Gap Decision Theory", Academic Press, 2001.

[24] F.M. Hemez, Y. Ben-Haim and S. Cogan, "Information-gap robustness for the test-analysis correlation of a non-linear transient simulator", Los Alamos National Laboratory Report LA-UR-02-3538, 2002.

DOI: 10.2514/6.2002-5420

[25] G.J. Klir and R.M. Smith, "On measuring uncertainty and uncertainty-based information", Annals of Mathematics, 2001.

[26] G. Choquet, "Theory of capacities", Annales de L'Institut Fourier, 1953, 54, 132-295.

[27] C.S. Reese, A.G. Wilson, M.S. Hamada, H.F. Martz and K.J. Ryan, "Integrated analysis of computer and physical experiments", Los Alamos National Laboratory Report LA-UR-00-2915, 2000.

[28] T.R. Bement, J.M. Booker, K.F. Sellers and N.D. Singpurwalla, "Membership functions and probability measures of fuzzy sets", Los Alamos National Laboratory Report LA-UR-00-3660, 2000.

[29] L.A. Bergman, "Uncertainty modelling in dynamical systems: a perspective", In Structural Dynamics @ 2000: Current Status and Future Directions. Ed. D.J. Ewins and D.J. Inman, Research Studies Press, Baldock, England, 2001, 9-16.

[30] P. Hayton, S. Utete, D. King, S. King, P. Anuzis and L. Tarassenko, "Static and dynamic novelty detection methods for jet engine health monitoring", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2007, 365, 593-514.

DOI: 10.1098/rsta.2006.1931

[31] K. Worden and G. Manson, "The application of machine learning to structural health monitoring", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2007, 365, 515-537.

DOI: 10.1098/rsta.2006.1938

[32] G. Manson, K. Worden and D.J. Allman, "Experimental validation of a structural health monitoring methodology. Part III. Damage location on an aircraft wing", Journal of Sound and Vibration, 2003, 259, 365-385.

DOI: 10.1006/jsvi.2002.5169

[33] K. Worden, G. Manson and N.R. Fieller, "Damage detection using outlier analysis", Journal of Sound and Vibration, 2000, 229, 647-667.

DOI: 10.1006/jsvi.1999.2514

[34] G. Shafer and R. Logan, "Implementing Dempster's rule for hierarchical evidence", Artificial Intelligence, 1987, 33, 271-298.

DOI: 10.1016/0004-3702(87)90040-3

[35] Worden K, Manson G, Denoeux T., "An evidence-based approach to damage localisation on an aircraft structure", Submitted to Mechanical Systems and Signal Processing 2007, 23, 1792-1804.

DOI: 10.1016/j.ymssp.2008.11.003

[36] T. Denoeux, "A neural network classifier based on Dempster-Shafer theory", IEEE Transactions on Systems, Man and Cybernetics, 2000, 30, 131-150.

DOI: 10.1109/3468.833094

[37] K. Worden, G. Manson, G. Hilson and S.G. Pierce, "Genetic optimisation of a neural damage locator" Journal of Sound and Vibration, 2008, 309, 529-544.

DOI: 10.1016/j.jsv.2007.07.035

[38] L. Tarassenko, "A Guide to Neural Computing Applications", Arnold, 1998.

[39] C.M. Bishop, "Neural Networks for Pattern Recognition", Oxford University Press, 1995.