Detection of Nonlinearity in Structures via Hilbert Transform Iran University of Science & Technology Jun 2009 Detection of Nonlinearity in Structures via Hilbert Transform Supervisor: Hamid Ahmadian BY: Mohammad Kalami Yazdi Http://Webpages.iust.ac.ir/Kalami Aerospace Division, Department of Mechanical Engineering

Contents Detection of nonlinearity Literature review Hilbert Transform Typical sources of nonlinearities Other Applications of Hilbert Transform Summary

Who is Hilbert ? David Hilbert (1862-1943) German mathematician Recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered or developed a broad range of fundamental ideas in many areas.

Who is Pioneer? Professor G R Tomlinson Pro Vice Chancellor for Research Director of the Rolls-Royce UTC Address: Department of Mechanical Engineering The University of Sheffield Mappin Street, Sheffield, S1 3JD Telephone: +44 (0) 114 222 7705 Fax: +44 (0) 114 222 7890 Email : g.tomlinson@sheffield.ac.uk

Typical Sources of Nonlinearities are: Geometric nonlinearity Inertia nonlinearity A nonlinear material behaviour Damping dissipation Boundary conditions typical nonlinearities include backlash and friction in control surfaces and joints, Nonlinearity may also arise in a damaged structure: fatigue cracks, rivets and bolts that subsequently open and close under dynamic loading or internal parts impacting upon each other.

Importance of Nonlinearity Detection :

Literature Review: M. Simon, G.R. Tomlinson, “Use of the Hilbert transform in modal analysis of linear and non-linear structures”, Journal of Sound and Vibration 96 (1984) 421–436 G.R Tomlinson, I Ahmed, “Hilbert transform procedure for detecting & quantifying nonlinearity in model testing”. Meccanica 22 (1987), pp. 123–132. G.R. Tomlinson, “Developments in the use of the Hilbert transform for detecting and quantifying non-linearity in frequency response functions”. Mechanical Systems and Signal Processing, I (2), 1987, 151- 173.

Literature Review: D. Spina, C. Valente, G.R. Tomlinson, “A new procedure for detecting nonlinearity from transient data using Gabor transform”, Nonlinear Dynamics 11 (1996) 235–254 K. Worden, G.R. Tomlinson, “ Nonlinearity in Structural Dynamics: Detection, Identification and Modelling ”, Institute of Physics Publishing, Bristol and Philadelphia, 2001. K.Worden , G.R.Tomlinson. The high-frequency behaviour of frequency response functions and its effect on their Hilbert transform. Proc. of 8th IMAC, 1990, Florida, pp.121-127.

Other Detection Techniques are: Harmonic Detection function (Auweraer et al., 1984 ) Inverse receptance method (He and Ewins, 1987 ) Complex stiffness method (Mertens et al., 1989 ) Gabor Transform (Spina et al, 1996 ) Wavelet Transform(Staszewski, 2000 ) Non-causal power ratio (Kim and Park, 1993 ) Backbone curve (Feldman, 1994) Carpet plots (Ewins, 2000 ) Autocorrelation functions of Residuals (Adams et al, 2000) Multisine excitations (Vanhoenacker et al., 2001) Nearest neighbour approach (Trendafilova et al., 2000 )

Hilbert Transform

Hilbert Transform for Causal Functions ( ):

Hilbert Transform : Hilbert Transform

Hilbert Transform for a causal system: non-causal system:

Hilbert Transform Detects Nonlinearity:

Detection of Hardening Cubic Stiffness:

Case Study

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Summary The Hilbert transform is a fast and effective means of testing for nonlinearity on the basis of a measured FRF. It has advantage over other methods, for example, in that it can be applied to a single FRF measured at a single level of excitation (as long as the nonlinearity has been adequately excited). Computation is fairly straightforward for a baseband FRF, but complications can arise for zoomed FRFs. However, the problems can be circumvented by the use of correction terms or a pole-zero computation.

Iran University of Science & Technology Jun 2009 The End Thank You for Your Kind Attention! Questions? “As you set out for Ithaka, Hope the journey is a long one, full of Adventure, full of Discovery. Laistrygonians and Cyclops, Angry Poseidon - Don't be afraid of them: You'll Never find things like that on your way as long as you keep your thoughts raised high, as long as a rare excitement, stirs your spirit and your body. Laistrygonians and Cyclops, wild Poseidon - you won't encounter them, unless you bring them along inside your soul, unless your soul sets them up in front of you.” Constantine Cavafy