1. Uncertainty, Variability and the Dynamics of Built-Up Structures Brian Mace brm@isvr.soton.ac.uk

2. Overview  The main problem: predicting response variability at acceptable computational cost  Mean, max, min response, worst-case behaviour etc, without 1000’s of repeated vibration analyses  Introductory comments  Examples of measured variability  High and mid frequency issues  Low frequencies  Component Mode Synthesis (CMS) and uncertainty  Concluding remarks

3. Introductory comments

4. Parameter variability  Variability from one structure to another  Manufacturing variations, dimensions, tolerances etc  Material properties, elastic modulus  Residual stresses  Variability of one structure with time  Environment (e.g. temperature)  Wear, fatigue etc  Operating conditions (load, orientation etc)  Variability in the analysis or measurement process  Repeatability and reproducibility etc  Modelling approximations or assumptions

5. Uncertainty  We don’t know the exact values of the parameters  Physical and geometric parameters  Density  Elastic modulus  Dimensions  Joint properties (stiffness etc)  Equations of motion  Assumptions  Approximations

6. Approaches to Modelling with Variability  Probabilistic  Data: probability density function, mean, variance etc  Realistic, but much information needed  Possibilistic  Data: range of possible values (min…max)  Simple, but often very conservative  High levels of uncertainty, high frequencies  Data: large uncertainty; universal statistics?  Statistical Energy Analysis (SEA); Energy Influence Coefficients  Problem: cost of multiple analyses (e.g. 100’s of FEAs)

7. Probabilistic methods  Monte Carlo Simulation (MCS)  Repeat full analysis many times with different input data  Generate statistics of output (response quantities)  Cost: high, even with efficient methods (sampling methods)  Stochastic FE  Perturbation methods  Sensitivities etc (e.g. variation of eigenvalues with a parameter   Reliability methods (FORM, SORM)

8. Possibilistic methods  Parameters and response lie in possible ranges  Conservative, can be very conservative min max  Fuzzy arithmetic/fuzzy FE  Interval analysis  “Fuzzy” membership  Only a possible range  Cost can be expensive:  Transformation methods  Vertex, corner etc

9. Examples of measured variability

10. Vehicle FRF variability  Isuzu Rodeo (98 vehicles)  Structure-borne and air-borne FRFs  Gaussian distribution generally “good” fit to response  Lognormal fit significantly poorer  Std dev ~ 40-50% Kompella and Bernhard (1996) Hills et al (2004)

11. Rodeo structure-borne FRF Rodeo Structure-borne 52.5Hz 375Hz Gaussian

12. Quantification of uncertainty: alloys wheels    

13. Environmental Variability: Natural Frequencies of a Bridge  Range ~ 6% with temperature  Range ~ 2% at given temperature (Cornwell et al, Expt Tech (1995) Alamosa canyon bridge Bridge deck temp. differential (F) Frequency (Hz)  Monitored to detect damage  Nat freq correlates to temperature differential and time-of-day

14. Variability and Spatial Correlation: Thickness of a CFRP Plate  Thickness varies with position  Clear spatial correlation  Thickness distribution Gaussian Cumulative distribution function of thickness: measured and Gaussian distribution Thickness as a function of (x,y) position in plate (Zehn and Seitov) Thickness

15. Variability and Spatial Correlation: Natural frequencies of CFRP Plate  Many FE realisations gives range ~ 5%  Assuming no spatial correlation gives predictions 5-7% higher  Spatial correlation is important (Zehn and Saitov) Natural frequencies of 50 FE realisations including spatial correlation of thickness: percentage deviation from mean

16. High and mid frequency issues

17. High frequencies  Subsystems have many modes  Large amounts of uncertainty  Asymptotic behaviour of mode statistics for large uncertainty P coup P in  Energy methods  Input power, coupling power, subsystem energy  Statistical energy analysis (SEA)  Energy influence coefficients (EICs)

18. Mid-frequencies: coupling stiff and flexible subsystems  E.g. beam-plate system  Beam:  Deterministic  Long wavelength, few modes  Model using FE  Plate:  Stochastic  Short wavelength, many modes  Model statistically (energy, SEA)  No single analysis method suitable

19. Typical example: beam-plate system input mobility Frequency (Hz) Input mobility Comments Plate adds damping and mass to beam modes Resonance peaks occur at lower freqs Fluctuations due to uncertain plate modes (Calculated using modal approach of Lin)

20. Models of the receiver  Diffuse wavefield  Modes, described statistically

21. Mid-frequency methods  Push FEA to the limit? Cost? Understanding?  Model reduction: CMS, AMLS etc  EICs  Waves  “Locally reacting impedance”  Statistical models, diffuse wavefield  Modal methods e.g.:  “Resound” 1  Interface decomposition 2  Fuzzy structures 3-5  Hybrid FE/SEA approaches  “Hybrid Resound” (Langley and Shorter) – PAM VA1 e.g. 1 Langley et al; 2 Ji Lin et al; 3 Soize; 4 Pierce et al; 5 Strasberg et al

22. CMS and uncertainty: low frequency applications

23. The CMS philosophy  Model reduction  System = individual components, coupled together  Solve more, smaller, “local” problems, 1 for each subsystem  Assemble and solve the “global” problem  Craig-Bampton (Fixed interface) CMS  Substructures: normal modes of vibration with interface fully fixed  Constraint modes: static deformation of the component under unit displacement of one interface DOF  DOFs = normal mode and constraint mode amplitudes

24. FEA of coupled subsystems: Structure of the mass and stiffness matrices 1 2 Coupling DOFs Interior DOFs, component 2 Interior DOFs, component 1 Physical DOFs  Size of global eigenvalue problem much smaller (reduced cost for repeated solutions) Component modal DOFs 0 0 Coupling DOFs Modes of component 2 Modes of component 1 diag Coupling stuff here Mass matrix only has coupling stuff here

25. CMS: the “baseline” analysis  Define component mass and stiffness matrices  Find subsystem modes and constraint modes  Assemble and solve for global modes  Cost mainly from  Subsystem modal analyses  Global modal analysis physical properties component modal properties FRF global modal properties cheap expensive CMS modal sum global modal analysis expensive

26. CMS and uncertainty modelling  Substructures and joints often statistically independent  Different manufacturing processes  Reduced size of model  Computational cost reduced  Reduced cost for repeated solution e.g. Monte Carlo Simulation  Reduced amount of data  Reanalysis only needed for uncertain substructures or joints  Issues:  Uncertainty quantification  Uncertainty propagation

27. CMS: uncertainty quantification physical properties component modal properties FRF global modal properties subsystem modal analyses modal sum global modal analysis  Response variation  Physical properties  Random fields, spatial correlation?  Experimental quantification expensive?  Component modal properties  Eigenvalues (discrete)  Eigenvectors?  CC modes?  Correlations?  Quantification: hammer test?

28. Propagation schemes Component modal Component physical Global response Global modal  Reduced size  MCS feasible?  Statistically independent  Only reanalyse substructures with significant uncertainty  Different propagation schemes for different components, e.g.  Perturbation  Full MCS  Hybrid probabilistic/possibilistic  Propagation schemes: analytical, MCS, perturbation, interval analysis etc

29. Perturbations and CMS  Perturbation methods very cheap  Derivative of eigenvalue i with respect to uncertain parameter p j :  Expressions for eigenvector sensitivity well known  From physical to component modal properties  p = vector of correlated FE properties  From component modal to global modal properties  p = vector of component modal properties  Sensitivities already known

30. CMS and uncertainty: perturbational approach  Introduce uncertainty into component modal properties CMS physical properties local modal properties FRF global modal properties CMS variability cheap very cheap perturbation response statistics  Local Modal Perturbational approach  Perturbation between local modal and global modal properties  MCS

31. FRF statistics baseline mobility; 20 realisations mean; 10/90 percentiles 051015202530 35 40 10 -2 10 0 2 Frequency  3-plate structure  Predictions:  Baseline response  Statistics of the response  LMP method:  Cost of statistics less than cost of baseline analysis

32. Concluding remarks

33.  Uncertainty and variability in properties and hence response  The main problem: predicting response variability at acceptable computational cost  Probabilistic and possibilistic analysis methods  Examples of measured variability – problem of quantifying variability  High and mid frequency issues  Low frequencies: CMS and uncertainty  Attractive framework for built-up structures  Reduced size of model and data  Perturbational approach based on CMS