worden.ppt

7/29/2019 Worden.ppt

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Where Non-Smooth Systems

Appear in Structural Dynamics

Keith Worden

Dynamics Research GroupDepartment of Mechanical Engineering

University of Sheffield


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NonlinearityNonlinearity 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 MonitoringActive/passive control of vibrations

Control


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System IdentificationAutomotive damper

(shock absorber)

Designed to be

nonlinear.

Physical model

prohibitively complicated.Bilinear.


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System IDStandard SDOF system,

( ) ( ) ( )my h y f y x t

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 goodinitial 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:

2

1 2

1

( , , , , ) ( )N

i i

i

J m c k k d y y


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Excellent solution:

Derivative-free.

Avoids local minima.

No need to

differentiate/integrate

time data.

Directly optimises on

Model PredictedOutput as opposed to

One-step-ahead

predictions.


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HysteresisSystems with memory:

Bouc-Wen model is versatile.

( )

| | | |n nmy cy ky z x t

z y z z y Ay

Nonlinear in the parameters.

Unmeasured state z.

Can use GA again or Differential Evolution.


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HydromountContains viscoelastic elements.

Valves (like shock absorber)produce non-smooth

nonlinearity.


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Freudenberg Model1 2

3

2 1 2 2 2 3 4 4 1 3

3 4

4 1 2 4 4 4 2 3 3 4

4 5 4 3 3 4 6 1 3

7 3 3 3 3 3

8 2 4 9 1 3 10

( ) ( )

( | | ) | |

( | |) | | ( )| | ( sgn( ))

( ) ( )

t t

t

t

t t t t t

z z

z l z l z l z z l z z z

z z

z h z z z z z h z h z

h h z h z z h z z z h z h z h z

F h z z z h z z z h z k z


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FrictionVery significant for high-speed, high-accuracy

machining.

Need: Friction models,

Control strategies.

Most basic model is Coulomb friction:

( ) sgn( )cF y F y


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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 Model0 1 2

0

0

| |

( )

( )

| |1

LG

c s c

s

F z z y b

y z

z y s y

F F Fs y

y

v


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An Experiment


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Particle Damper


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Structural Health MonitoringRytters 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 AgainOften, 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 likeLiapunov 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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Correlation functions Force

Deformation

])(')('[)( 2''2 ixkixEk

xx


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Holder Exponent

Acceleration time-histories

Holder exponent (In-Axis)


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SDOF Model of Cracked Beam

Parameter

represents depth of

crack


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Bifurcation diagram for = 0.2.


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Problem is that system bifurcates and shifts in and outof chaos; features like liapunov exponents,correlation dimension etc. will not always work and

are not monotonically increasing with damageseverity.

Figure shows dependence on frequency, but same

picture appears with crack depth as independentvariable

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 methodsfor systems that bifurcate.


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Acknowledgements Lawrie Virgin (Duke University)

Chuck Farrar, Gyuhae Park (Los Alamos NationalLaboratory)

Farid Al Bender (KUL, Leuven)

Jem Rongong, Chian Wong, Brian Deacon, JonnyHaywood (University of Sheffield)

Andreas Kyprianou (University of Cyprus)

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