Embed Size (px)
Citation preview
IMAC XXV, Orlando, 19 February 2007
Experimental Case Studies onUncertainty Quantification in
Structural DynamicsS. Adhikari, K. Lonkar and M. I. Friswell
Department of Aerospace Engineering, University of Bristol, Bristol, U.K.
Email: [email protected]
URL: http://www.aer.bris.ac.uk/contact/academic/adhikari/home.html
Experimental UQ – p.1/30
IMAC XXV, Orlando, 19 February 2007
Outline of the presentation
Introduction
Probabilistic structural dynamics
Experimental case study 1: Fixed beam withrandomly placed masses
Experimental case study 2: Cantilever platewith randomly placed oscillators
Conclusions & discussions
Experimental UQ – p.2/30
IMAC XXV, Orlando, 19 February 2007
Overview of Predictive Methodsin Engineering
There are five key steps:
Physics (mechanics) model building
Uncertainty Quantification (UQ)
Uncertainty Propagation (UP)
Model Verification & Validation (V & V)
Prediction
Tools are available for each of these steps. Focus ofthis talk is mainly on UQ in linear dynamicalsystems.
Experimental UQ – p.3/30
IMAC XXV, Orlando, 19 February 2007
Why uncertainty?
Different sources of uncertainties in the modeling andsimulation of dynamic systems may be attributed, but notlimited, to the following factors:
Mathematical models: equations (linear, non-linear),geometry, damping model (viscous, non-viscous,fractional derivative), boundary conditions/initialconditions, input forces;
Model parameters: Young’s modulus, mass density,Poisson’s ratio, damping model parameters (dampingcoefficient, relaxation modulus, fractional derivativeorder)
Experimental UQ – p.4/30
IMAC XXV, Orlando, 19 February 2007
Why uncertainty?
Numerical algorithms: weak formulations, discretisationof displacement fields (in finite element method),discretisation of stochastic fields (in stochastic finiteelement method), approximate solution algorithms,truncation and roundoff errors, tolerances in theoptimization and iterative methods, artificial intelligent(AI) method (choice of neural networks)
Measurements: noise, resolution (number of sensors andactuators), experimental hardware, excitation method(nature of shakers and hammers), excitation andmeasurement point, data processing (amplification,number of data points, FFT), calibration
Experimental UQ – p.5/30
IMAC XXV, Orlando, 19 February 2007
Structural dynamics
The equation of motion:
Mx(t) + Cx(t) + Kx(t) = p(t)
Due to the presence of uncertainty M, C and Kbecome random matrices.
The main objectives in the ‘forward problem’are:
to quantify uncertainties in the systemmatricesto predict the variability in the responsevector x
Experimental UQ – p.6/30
IMAC XXV, Orlando, 19 February 2007
Current Methods
Two different approaches are currently available
Low frequency : Stochastic Finite ElementMethod (SFEM) - assumes that stochasticfields describing parametric uncertainties areknown in details
High frequency : Statistical Energy Analysis(SEA) - do not consider parametricuncertainties in details
Experimental UQ – p.7/30
IMAC XXV, Orlando, 19 February 2007
Experimental Study: Fixed beam
A fixed-fixed beam
Experimental UQ – p.8/30
IMAC XXV, Orlando, 19 February 2007
Beam properties
Beam Properties Numerical values
Length (L) 1200 mm
Width (b) 40.06 mm
Thickness (th) 2.05 mm
Mass density (ρ) 7800 Kg/m3
Young’s modulus (E) 2.0× 105 MPa
Mass per unit length (ρl) 0.641 Kg/m
Total weight 0.7687 Kg
Experimental UQ – p.9/30
IMAC XXV, Orlando, 19 February 2007
Randomly placed masses
12 randomly placed masses (magnets), each weighting 2 g (total variation: 3.2%): mass
locations are generated using uniform distribution
Experimental UQ – p.10/30
IMAC XXV, Orlando, 19 February 2007
Randomly placed masses
Mean (m) Standard deviation (m)
0.2709 0.0571
0.3390 0.0906
0.3972 0.1043
0.4590 0.1034
0.5215 0.1073
0.5769 0.1030
0.6398 0.1029
0.6979 0.1021
0.7544 0.0917
0.8140 0.0837
0.8757 0.0699
0.9387 0.0530
Gaussian distribution of mass locations along the beam
Experimental UQ – p.11/30
IMAC XXV, Orlando, 19 February 2007
Randomly placed masses
0 0.2 0.4 0.6 0.8 1 1.2
2
4
6
8
10
12
14
Length along the beam (m)
Sam
ple
num
ber
First 15 samples of the locations of 12 masses along the length of the beam.
Experimental UQ – p.12/30
IMAC XXV, Orlando, 19 February 2007
Impulse excitation using ashaker
The shaker used as an impulse hammer using SimulinkTM. A hard steel tip used.
Experimental UQ – p.13/30
IMAC XXV, Orlando, 19 February 2007
FRF Variability: completespectrum
Variability in the amplitude of the driving-point-FRF of the beam.
Experimental UQ – p.14/30
IMAC XXV, Orlando, 19 February 2007
FRF Variability: Low Freq
0 100 200 300 400 500 600 700 800 900 1000−80
−70
−60
−50
−40
−30
−20
−10
0
10
20
Frequency ω (Hz)
Log a
mplitu
de (d
B) of
point
1
Baseline systemEnsamble average5% points95% points100 random samples
Variability in the amplitude of the driving-point-FRF of the beam.
Experimental UQ – p.15/30
IMAC XXV, Orlando, 19 February 2007
FRF Variability: Mid Freq
1000 1500 2000 2500−80
−70
−60
−50
−40
−30
−20
−10
0
10
20
Frequency ω (Hz)
Log a
mplitu
de (d
B) of
point
1
Baseline systemEnsamble average5% points95% points100 random samples
Variability in the amplitude of the driving-point-FRF of the beam.
Experimental UQ – p.16/30
IMAC XXV, Orlando, 19 February 2007
FRF Variability: High Freq
2600 2800 3000 3200 3400 3600 3800 4000 4200−80
−70
−60
−50
−40
−30
−20
−10
0
10
20
Frequency ω (Hz)
Log a
mplitu
de (d
B) of
point
1
Baseline systemEnsamble average5% points95% points100 random samples
Variability in the amplitude of the driving-point-FRF of the beam.
Experimental UQ – p.17/30
IMAC XXV, Orlando, 19 February 2007
FRF Variability: completespectrum
Variability in the amplitude of a cross-FRF of the beam.
Experimental UQ – p.18/30
IMAC XXV, Orlando, 19 February 2007
FRF Variability: Low Freq
0 100 200 300 400 500 600 700 800 900 1000−80
−70
−60
−50
−40
−30
−20
−10
0
10
20
Frequency ω (Hz)
Log a
mplitu
de (d
B) of
point
3
Baseline systemEnsamble average5% points95% points100 random samples
Variability in the amplitude of a cross-FRF of the beam.
Experimental UQ – p.19/30
IMAC XXV, Orlando, 19 February 2007
FRF Variability: Mid Freq
1000 1500 2000 2500−80
−70
−60
−50
−40
−30
−20
−10
0
10
20
Frequency ω (Hz)
Log a
mplitu
de (d
B) of
point
3
Baseline systemEnsamble average5% points95% points100 random samples
Variability in the amplitude of a cross-FRF of the beam.
Experimental UQ – p.20/30
IMAC XXV, Orlando, 19 February 2007
FRF Variability: High Freq
2600 2800 3000 3200 3400 3600 3800 4000 4200−80
−70
−60
−50
−40
−30
−20
−10
0
10
20
Frequency ω (Hz)
Log a
mplitu
de (d
B) of
point
3
Baseline systemEnsamble average5% points95% points100 random samples
Variability in the amplitude of a cross-FRF of the beam.
Experimental UQ – p.21/30
IMAC XXV, Orlando, 19 February 2007
Experimental Study: cantileverplate
A cantilever plate: Length: 998 mm, Width: 530 mm, Thickness: 3 mm,
Density: 7860 kg/m3, Young’s Modulus: 200 GPa
Experimental UQ – p.22/30
IMAC XXV, Orlando, 19 February 2007
Unmodelled dynamics
10 randomly placed oscillator; oscillatory mass: 121.4 g, fixed mass: 2 g, spring stiffness vary
from 10 - 12 KN/m
Experimental UQ – p.23/30
IMAC XXV, Orlando, 19 February 2007
FRF Variability: completespectrum
Variability in the amplitude of a cross-FRF of the plate.
Experimental UQ – p.24/30
IMAC XXV, Orlando, 19 February 2007
FRF Variability: Low Freq
Variability in the amplitude of a cross-FRF of the plate.
Experimental UQ – p.25/30
IMAC XXV, Orlando, 19 February 2007
FRF Variability: Mid Freq
Variability in the amplitude of a cross-FRF of the plate.
Experimental UQ – p.26/30
IMAC XXV, Orlando, 19 February 2007
FRF Variability: High Freq
Variability in the amplitude of a cross-FRF of the plate.
Experimental UQ – p.27/30
IMAC XXV, Orlando, 19 February 2007
Conclusions
Experimental results involving stochastic dynamicalsystems is required for the uncertainty quantification andvalidation of numerical models of complex systems.
Two experimental studies are described which may beused for this purpose.
The fixed-fixed beam is easy to model and the results ofa 100 sample experiment with randomly placed masseswere described in this paper.
The cantilever plate is ‘perturbed’ by 10 randomly placedoscillators. Again, a 100-sample test is conducted andthe results are described.
Experimental UQ – p.28/30
IMAC XXV, Orlando, 19 February 2007
Conclusions
Special care has been taken so that the uncertainty in theresponse only arises from the randomness in the masslocations.
Statistics of the frequency response function measured atthree points of the beam were obtained for low, mediumand high frequency ranges.
It is expected that this data can be used for modelvalidation and uncertainty quantification of dynamicalsystems. Data presented here will be available in thewww.
Experimental UQ – p.29/30
IMAC XXV, Orlando, 19 February 2007
Open Issues
How one may validate/update a stochastic dynamicalmodel when experimental data of stochastic nature (suchas described here) is available?
What can be done about the limited sample size (oftenonly one!) in experimental results?
Experimental UQ – p.30/30
![Combined parametric–nonparametric uncertainty ...engweb.swan.ac.uk/~adhikaris/fulltext/journal/ft178.pdfapproaches combined with system reduction methods [15–18] and algebraic](https://img.dokumen.tips/doc/110x75/5f64147c1c7e351a7b79abce/combined-parametricaaoenonparametric-uncertainty-adhikarisfulltextjournalft178pdf.jpg)
