1. 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
Aerospace Division, Department of Mechanical Engineering

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

3. 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.

4. 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) Fax: +44 (0)

5. 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.

6. Importance of Nonlinearity Detection :

7. 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,

8. 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

9. 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 )

10. Hilbert Transform

11. Hilbert Transform
for Causal Functions ( ):

12. Hilbert Transform
: Hilbert Transform

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

14. Hilbert Transform Detects Nonlinearity:

15. Detection of Hardening Cubic Stiffness:

16. Case Study

17. Case Study
Data , Hilbert Transform

18. Data ----- , Hilbert Transform - - -
Case Study
Data , Hilbert Transform

19. Case Study
Data , Hilbert Transform

20. Case Study
Data , Hilbert Transform

21. 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.

22. Iran University of Science & Technology Jun 2009
The End
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