1. Where Non-Smooth Systems Appear in Structural Dynamics
Keith Worden
Dynamics Research Group
Department of Mechanical Engineering
University of Sheffield

2. Nonlinearity Nonlinearity is present in many engineering problems:
Demountable structures with clearances and friction.
Flexible structures – large amplitude motions.
Aeroelasticity – limit cycles.
Automobiles: squeaks and rattles, brake squeal, dampers.
Vibration isolation: viscoelastics, hysteresis.
Sensor/actuator nonlinearity: piezoelectrics…
In many cases, the nonlinearity is non-smooth. 2

3. So, where are the problems in Structural Dynamics?
System Identification
Structural Health Monitoring
Active/passive control of vibrations
Control…

4. System Identification
Automotive damper (shock absorber)
Designed to be nonlinear.
Physical model prohibitively complicated.
Bilinear.

5. System ID Standard SDOF system,
If nonlinearities are ‘linear in the parameters’ there are many powerful techniques available.
Even the most basic piecewise-linear system presents a problem.

6. Everything OK if we know d – linear in the parameters.
Otherwise need nonlinear least-squares.
Iterative - need good initial estimates.
Can use Genetic Algorithm.

7. Genetic Algorithm
Encode parameters as binary bit-string – Individuals.
Work with population of solutions.
Combine solutions via genetic operators:
Selection
Crossover
Mutation
Minimise cost function:

8. Excellent solution:
Derivative-free.
‘Avoids local minima’.
No need to differentiate/integrate time data.
Directly optimises on ‘Model Predicted Output’ as opposed to ‘One-step-ahead’ predictions.

9. Hysteresis Systems with ‘memory’: Bouc-Wen model is versatile.
Nonlinear in the parameters.
Unmeasured state z.
Can use GA again – or Differential Evolution.

10. Hydromount Contains viscoelastic elements.
Valves (like shock absorber) produce non-smooth nonlinearity.

11. Freudenberg Model

12. Friction Very significant for high-speed, high-accuracy machining.
Need:
Friction models,
Control strategies.
Most basic model is Coulomb friction:

13. Far too simplistic:
Static/dynamic friction.
Presliding/sliding regimes.
Stribeck effect…
Various models in use: white/grey/black.

14. Stribeck Curve

15. LuGre Model

16. An Experiment

18. Particle Damper

19. Structural Health Monitoring
Rytter’s hierarchy:
Detection
Location
Severity
Prognosis
Two main approaches:
Inverse problem
Pattern Recognition

20. Are These Systems Damaged?
Did you use pattern recognition?

21. Pattern Recognition: D2D
Data acquisition
Pre-processing
Feature extraction
Classification
Decision
Critical step is often Feature Extraction.

22. Dog or Cat

23. Nonlinearity Again
Often, the occurrence of damage will change the structure of interest from a linear system to a nonlinear system e.g. a ‘breathing’ crack.
This observation can be exploited in terms of selection of features, e.g. one can work with features like Liapunov exponents of time-series; if chaos is observed, system must be nonlinear. But…

24. Tests for Nonlinearity
Homogeneity
Reciprocity
Coherence
FRF distortion
Hilbert transform
Correlation functions

25. Correlation functions

26. Holder Exponent
Acceleration time-histories
Holder exponent (In-Axis)

27. SDOF Model of Cracked Beam
Parameter α ‘represents’ depth of crack

28. Bifurcation diagram for α = 0.2.

29. Problem is that system bifurcates and shifts in and out of chaos; features like liapunov exponents, correlation dimension etc. will not always work and are not monotonically increasing with damage severity.
Figure shows dependence on frequency, but same picture appears with ‘crack depth’ as independent variable
Are there better features?

30. Rocking (Thanks to Lawrie Virgin)

33. What needs to be done?
Development of signal processing tools like estimator of Holder exponent.
Better friction models (white/grey/black).
Parameter estimation/optimisation methods (as a side-issue, convergence results for GAs etc.)
Control methods for non-smooth systems.
Versatile hysteresis models.
Understanding of high-dimensional nonlinear models (e.g. FE).

34. Quantities that increase monotonically with ‘severity of nonlinearity’?
Engineers like random excitation - tools for stochastic DEs and PDEs with non-smooth nonlinearities.
Contact/friction models for DEM.
Sensitivity analysis/uncertainty propagation methods for systems that bifurcate.

35. Acknowledgements Lawrie Virgin (Duke University)
Chuck Farrar, Gyuhae Park (Los Alamos National Laboratory)
Farid Al Bender (KUL, Leuven)
Jem Rongong, Chian Wong, Brian Deacon, Jonny Haywood (University of Sheffield)
Andreas Kyprianou (University of Cyprus)