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 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 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 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 order) Experimental UQ – p.4/30 IMAC XXV, Orlando, 19 February 2007
Why uncertainty? Numerical algorithms: weak formulations, discretisation of 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 optimization 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 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 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 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 ( t h ) 2.05 mm 7800 Kg/m 3 Mass density ( ρ ) 2 . 0 × 10 5 MPa Young’s modulus ( E ) Mass per unit length ( ρ l ) 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 14 12 10 Sample number 8 6 4 2 0 0.2 0.4 0.6 0.8 1 1.2 Length along the beam (m) 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 a shaker The shaker used as an impulse hammer using Simulink TM . A hard steel tip used. Experimental UQ – p.13/30 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 IMAC XXV, Orlando, 19 February 2007
FRF Variability: Low Freq 20 Baseline system Ensamble average 10 5% points 95% points 0 100 random samples Log amplitude (dB) of point 1 −10 −20 −30 −40 −50 −60 −70 −80 0 100 200 300 400 500 600 700 800 900 1000 Frequency ω (Hz) 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 20 Baseline system Ensamble average 10 5% points 95% points 0 100 random samples Log amplitude (dB) of point 1 −10 −20 −30 −40 −50 −60 −70 −80 1000 1500 2000 2500 Frequency ω (Hz) 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 20 Baseline system Ensamble average 10 5% points 95% points 0 100 random samples Log amplitude (dB) of point 1 −10 −20 −30 −40 −50 −60 −70 −80 2600 2800 3000 3200 3400 3600 3800 4000 4200 Frequency ω (Hz) 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: complete spectrum 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 20 Baseline system Ensamble average 10 5% points 95% points 0 100 random samples Log amplitude (dB) of point 3 −10 −20 −30 −40 −50 −60 −70 −80 0 100 200 300 400 500 600 700 800 900 1000 Frequency ω (Hz) 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 20 Baseline system Ensamble average 10 5% points 95% points 0 100 random samples Log amplitude (dB) of point 3 −10 −20 −30 −40 −50 −60 −70 −80 1000 1500 2000 2500 Frequency ω (Hz) 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 20 Baseline system Ensamble average 10 5% points 95% points 0 100 random samples Log amplitude (dB) of point 3 −10 −20 −30 −40 −50 −60 −70 −80 2600 2800 3000 3200 3400 3600 3800 4000 4200 Frequency ω (Hz) Variability in the amplitude of a cross-FRF of the beam. Experimental UQ – p.21/30 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 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: complete spectrum 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 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 oscillators. Again, a 100-sample test is conducted and the 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 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 IMAC XXV, Orlando, 19 February 2007
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