Where Non-Smooth Systems Appear in Structural Dynamics

Keith WordenDynamics Research GroupDepartment of Mechanical EngineeringUniversity of Sheffield

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.

So, where are the problems in Structural Dynamics?

System IdentificationStructural Health MonitoringActive/passive control of vibrationsControl…

System IdentificationAutomotive damper (shock absorber)

Designed to be nonlinear.

Physical model prohibitively complicated.

Bilinear.

System ID

Standard 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.

Everything OK if we know d – linear in the parameters.

Otherwise need nonlinear least-squares.

Iterative - need good initial estimates.

Can use Genetic Algorithm.

Genetic AlgorithmEncode parameters as binary

bit-string – Individuals.Work with population of

solutions.Combine solutions via genetic

operators:

Selection

Crossover

MutationMinimise cost function:

21 2

1

ˆ( , , , , ) ( )N

i ii

J m c k k d y y

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.

HysteresisSystems with ‘memory’:

Bouc-Wen model is versatile.

( )

| | | |n n

my 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.

Hydromount

Contains viscoelastic elements.

Valves (like shock absorber) produce non-smooth nonlinearity.

Freudenberg Model1 2

32 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

FrictionVery significant for high-speed, high-accuracy

machining.

Need:Friction models,Control strategies.

Most basic model is Coulomb friction:

( ) sgn( )cF y F y

Far too simplistic:Static/dynamic friction.Presliding/sliding regimes.Stribeck effect…

Various models in use: white/grey/black.

Stribeck Curve

LuGre Model0 1 2

0

0

| |

( )

( )| |

1

LG

c s c

s

F z z y b

y zz y

s y

F F Fs y

yv

An Experiment

Particle Damper

Structural Health MonitoringRytter’s hierarchy:DetectionLocationSeverityPrognosis

Two main approaches:Inverse problemPattern Recognition

Are These Systems Damaged?

Did you use pattern recognition?

Pattern Recognition: D2DData acquisitionPre-processingFeature extractionClassificationDecision

Critical step is often Feature Extraction.

Dog or Cat

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…

Tests for NonlinearityHomogeneityReciprocityCoherenceFRF distortionHilbert transformCorrelation functions

Correlation functions

Force

Deformation

])(')('[)( 2

'' 2 ixkixEkxx

Holder Exponent

Acceleration time-histories

Holder exponent (In-Axis)

SDOF Model of Cracked Beam

Parameter α ‘represents’ depth of crack

Bifurcation diagram for α = 0.2.

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?

Rocking (Thanks to Lawrie Virgin)

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).

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.

AcknowledgementsLawrie 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)