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

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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.

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

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

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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.

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

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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:

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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.

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

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

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Freudenberg Model

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

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

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Stribeck Curve

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LuGre Model

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An Experiment

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Particle Damper

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

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Are These Systems Damaged? Did you use pattern recognition?

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

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Dog or Cat

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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…

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

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

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

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Bifurcation diagram for α = 0.2.

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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?

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Rocking (Thanks to Lawrie Virgin)

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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).

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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.

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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)

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