Experimental Case Studies on Uncertainty Quantification in - - PowerPoint PPT Presentation

▶

Feb 24, 2024 542 likes •865 views

Experimental Case Studies on Uncertainty Quantification in Structural Dynamics S. Adhikari, K. Lonkar and M. I. Friswell Department of Aerospace Engineering, University of Bristol, Bristol, U.K. Email: S.Adhikari@bristol.ac.uk URL:

SLIDE 1

IMAC XXV, Orlando, 19 February 2007

Experimental Case Studies on Uncertainty Quantification in Structural Dynamics

S. Adhikari, K. Lonkar and M. I. Friswell

Department of Aerospace Engineering, University of Bristol, Bristol, U.K. Email: S.Adhikari@bristol.ac.uk URL: http://www.aer.bris.ac.uk/contact/academic/adhikari/home.html

Experimental UQ – p.1/30

SLIDE 2

IMAC XXV, Orlando, 19 February 2007

Outline of the presentation

Introduction Probabilistic structural dynamics Experimental case study 1: Fixed beam with randomly placed masses Experimental case study 2: Cantilever plate with randomly placed oscillators Conclusions & discussions

Experimental UQ – p.2/30

SLIDE 3

IMAC XXV, Orlando, 19 February 2007

Overview of Predictive Methods in 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 of this talk is mainly on UQ in linear dynamical systems.

Experimental UQ – p.3/30

SLIDE 4

IMAC XXV, Orlando, 19 February 2007

Why uncertainty?

Different sources of uncertainties in the modeling and simulation of dynamic systems may be attributed, but not limited, to the following factors: Mathematical models: equations (linear, non-linear), geometry, damping model (viscous, non-viscous, fractional derivative), boundary conditions/initial conditions, input forces; Model parameters: Young’s modulus, mass density, Poisson’s ratio, damping model parameters (damping coefficient, relaxation modulus, fractional derivative

rder)

Experimental UQ – p.4/30

SLIDE 5

IMAC XXV, Orlando, 19 February 2007

Why uncertainty?

Numerical algorithms: weak formulations, discretisation

f displacement fields (in finite element method),

discretisation of stochastic fields (in stochastic finite element method), approximate solution algorithms, truncation and roundoff errors, tolerances in the

ptimization and iterative methods, artificial intelligent

(AI) method (choice of neural networks) Measurements: noise, resolution (number of sensors and actuators), experimental hardware, excitation method (nature of shakers and hammers), excitation and measurement point, data processing (amplification, number of data points, FFT), calibration

Experimental UQ – p.5/30

SLIDE 6

IMAC XXV, Orlando, 19 February 2007

Structural dynamics

The equation of motion: M¨ x(t) + C ˙ x(t) + Kx(t) = p(t) Due to the presence of uncertainty M, C and K become random matrices. The main objectives in the ‘forward problem’ are: to quantify uncertainties in the system matrices to predict the variability in the response vector x

Experimental UQ – p.6/30

SLIDE 7

IMAC XXV, Orlando, 19 February 2007

Current Methods

Two different approaches are currently available Low frequency : Stochastic Finite Element Method (SFEM) - assumes that stochastic fields describing parametric uncertainties are known in details High frequency : Statistical Energy Analysis (SEA) - do not consider parametric uncertainties in details

Experimental UQ – p.7/30

SLIDE 8

IMAC XXV, Orlando, 19 February 2007

Experimental Study: Fixed beam

A fixed-fixed beam

Experimental UQ – p.8/30

SLIDE 9

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

SLIDE 10

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

SLIDE 11

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

SLIDE 12

IMAC XXV, Orlando, 19 February 2007

Randomly placed masses

0.2 0.4 0.6 0.8 1 1.2 2 4 6 8 10 12 14 Length along the beam (m) Sample number

First 15 samples of the locations of 12 masses along the length of the beam.

Experimental UQ – p.12/30

SLIDE 13

IMAC XXV, Orlando, 19 February 2007

Impulse excitation using a shaker

The shaker used as an impulse hammer using SimulinkTM. A hard steel tip used.

Experimental UQ – p.13/30

SLIDE 14

IMAC XXV, Orlando, 19 February 2007

FRF Variability: complete spectrum

Variability in the amplitude of the driving-point-FRF of the beam.

Experimental UQ – p.14/30

SLIDE 15

IMAC XXV, Orlando, 19 February 2007

FRF Variability: Low Freq

100 200 300 400 500 600 700 800 900 1000 −80 −70 −60 −50 −40 −30 −20 −10 10 20 Frequency ω (Hz) Log amplitude (dB) of point 1 Baseline system Ensamble average 5% points 95% points 100 random samples

Variability in the amplitude of the driving-point-FRF of the beam.

Experimental UQ – p.15/30

SLIDE 16

IMAC XXV, Orlando, 19 February 2007

FRF Variability: Mid Freq

1000 1500 2000 2500 −80 −70 −60 −50 −40 −30 −20 −10 10 20 Frequency ω (Hz) Log amplitude (dB) of point 1 Baseline system Ensamble average 5% points 95% points 100 random samples

Variability in the amplitude of the driving-point-FRF of the beam.

Experimental UQ – p.16/30

SLIDE 17

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 10 20 Frequency ω (Hz) Log amplitude (dB) of point 1 Baseline system Ensamble average 5% points 95% points 100 random samples

Variability in the amplitude of the driving-point-FRF of the beam.

Experimental UQ – p.17/30

SLIDE 18

IMAC XXV, Orlando, 19 February 2007

FRF Variability: complete spectrum

Variability in the amplitude of a cross-FRF of the beam.

Experimental UQ – p.18/30

SLIDE 19

IMAC XXV, Orlando, 19 February 2007

FRF Variability: Low Freq

100 200 300 400 500 600 700 800 900 1000 −80 −70 −60 −50 −40 −30 −20 −10 10 20 Frequency ω (Hz) Log amplitude (dB) of point 3 Baseline system Ensamble average 5% points 95% points 100 random samples

Variability in the amplitude of a cross-FRF of the beam.

Experimental UQ – p.19/30

SLIDE 20

IMAC XXV, Orlando, 19 February 2007

FRF Variability: Mid Freq

1000 1500 2000 2500 −80 −70 −60 −50 −40 −30 −20 −10 10 20 Frequency ω (Hz) Log amplitude (dB) of point 3 Baseline system Ensamble average 5% points 95% points 100 random samples

Variability in the amplitude of a cross-FRF of the beam.

Experimental UQ – p.20/30

SLIDE 21

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 10 20 Frequency ω (Hz) Log amplitude (dB) of point 3 Baseline system Ensamble average 5% points 95% points 100 random samples

Variability in the amplitude of a cross-FRF of the beam.

Experimental UQ – p.21/30

SLIDE 22

IMAC XXV, Orlando, 19 February 2007

Experimental Study: cantilever plate

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

SLIDE 23

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

SLIDE 24

IMAC XXV, Orlando, 19 February 2007

FRF Variability: complete spectrum

Variability in the amplitude of a cross-FRF of the plate.

Experimental UQ – p.24/30

SLIDE 25

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

SLIDE 26

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

SLIDE 27

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

SLIDE 28

IMAC XXV, Orlando, 19 February 2007

Conclusions

Experimental results involving stochastic dynamical systems is required for the uncertainty quantification and validation of numerical models of complex systems. Two experimental studies are described which may be used for this purpose. The fixed-fixed beam is easy to model and the results of a 100 sample experiment with randomly placed masses were described in this paper. The cantilever plate is ‘perturbed’ by 10 randomly placed

scillators. Again, a 100-sample test is conducted and

the results are described.

Experimental UQ – p.28/30

SLIDE 29

IMAC XXV, Orlando, 19 February 2007

Conclusions

Special care has been taken so that the uncertainty in the response only arises from the randomness in the mass locations. Statistics of the frequency response function measured at three points of the beam were obtained for low, medium and high frequency ranges. It is expected that this data can be used for model validation and uncertainty quantification of dynamical

systems. Data presented here will be available in the

www.

Experimental UQ – p.29/30

SLIDE 30

IMAC XXV, Orlando, 19 February 2007

Open Issues

How one may validate/update a stochastic dynamical model when experimental data of stochastic nature (such as described here) is available? What can be done about the limited sample size (often

nly one!) in experimental results?

Experimental UQ – p.30/30