1. OMAC 2008: 1 Operational Modal Analysis Conference 2008 Modal testing of structures under operational conditions. Real Boundary Conditions Real Operational Loads

2. OMAC 2008: 2 OMAC 2008 Schedule 09:00 – 10:40 Technical Session 1 – Introduction to OMA What is Modal Analysis Traditional Modal Analysis versus Operational Modal Analysis Applications of OMA – Some Case Studies 11:10 - 13:00 Technical Session 2 – Analysis of a Real Structure Preliminary (Automatic) Modal Analysis The Frequency Domain Decomposition (FDD) Technique The Stochastic Subspace Identification Technique. Keynote 14:15 – 15:15 Technical Session 3 – Testing in Practice Generating Test Geometry and Plan the Test Testing using Reference Sensors and Moving Sensors 15:35 – 16:30 Technical Session 4 – Rounding Off Last Questions and Answers

3. OMAC 2008: 3 What is Modal Analysis – Modal Information One of the few things in life that will never go out of fashion: The Newton’s 2 nd Law of Motion f = M a Force Mass Acceleration Modal Information provides a systematic and decoupled way of describing how a structure responds when forces are applied to it. Sir Isaac Newton (1642-1727)

4. OMAC 2008: 4 What is Modal Analysis – Modal Information What does modal information mean ? Excitation Response Force Motion Input Output H(  ) = == The Frequency Response Function Modal Decomposed Frequency Response Function

5. OMAC 2008: 5 What is Modal Analysis – What are Modes? 1 st Mode Shape 2 nd Mode Shape

6. OMAC 2008: 6 What is Modal Analysis – What are Modes? Damping Frequency Frequency Domain Time Domain

7. OMAC 2008: 7 Traditional Modal Technology Input Output Time DomainFrequency Domain FFT Inverse FFT Excitation Response Force Motion Input Output H(  ) = == Frequency Response Function Impulse Response Function

8. OMAC 2008: 8 Shaker excitation Small homogenous structures Quick Polyreference technique Fast method - no fixtures required Out In Out In Multichannel response or response points may be moved Large or complex structures Various excitation signals possible Time consuming - installation work to be done Hammer excitation Excitation moved Traditional Modal Technology

9. OMAC 2008: 9 Traditional Modal Technology - Limitations No external excitation during testing – Test Rig Required! Improper boundaries and excitation levels have to be accepted sometimes. Hammers and shakers limits applications: Modes of symmetric structures are difficult to find due to the single input. Large structures impossible to excite artificially.

10. OMAC 2008: 10 Determination of Modal Model by response testing only –No measurement of input forces required –Measurement procedure similar to Operational Deflection Shapes (ODS) Determination of Modal Model in-situ under operational conditions –True boundary conditions. –Correct excitation level giving correct Modal Model in case of amplitude dependent non-linearities. Used in Civil Engineering applications –Bridges and buildings –Off-shore platforms etc. Used in Mechanical Engineering applications –On-road and in-flight testing etc. –Rotating Machinery Operational Modal Analysis – (OMA)

11. OMAC 2008: 11 Operational Modal Analysis - Procedure Determination of Modal parameters based on natural excitation Measurement of responses in a number of DOF’s –simultaneously –by roving accelerometers with one or more fixed accelerometers as references Fixed Reference Accelerometers Accelerometers are moved for each data set

12. OMAC 2008: 12 Measured Responses Stationary Zero Mean Gaussian White Noise Model of the combined system is estimated from measured responses Excitation Filter (linear, time-invariant) Structural System (linear, time-invariant) Unknown excitation forces Combined System Modal Model of Structural System extracted from estimated model of Combined System Operational Modal Analysis – Combined Model

13. OMAC 2008: 13 If the system is excited by white noise the output spectrum contains full information of the structure as all modes are excited equally But this is in general not the case! Structural System Force Spectrum Output Spectrum Operational Modal Analysis Operational Modal Analysis – Combined Model

14. OMAC 2008: 14 In general the excitation has a spectral distribution Modes are weighted by the spectral distribution of the input force Both the “modes” originating from the excitation signal and the structural modes are observed as “modes” in the Response Structural System Force Spectrum Combined Spectrum Operational Modal Analysis Operational Modal Analysis – Combined Model

15. OMAC 2008: 15 Noise also contributes to the Response Measurement Noise Computational Noise Force Spectrum Structural System Combined Spectrum Operational Modal Analysis Operational Modal Analysis – Combined Model

16. OMAC 2008: 16 Operational Modal Analysis – Combined Model Rotating parts creates Harmonic vibrations Measurement Noise Computational Noise Force Spectrum The “Modes” in the combined spectrum contains information of The system under test (Physical Modes) Input Force(Non-physical “Modes”) Noise(Non-physical “Modes”) Harmonics(Non-physical “Modes”) Rotating Parts Structural System Combined Spectrum Operational Modal Analysis

17. OMAC 2008: 17 Operational Modal Analysis – Advantages OMA is MIMO whereas traditional modal testing in general is SISO or SIMO. In case of a symmetric structures with closely spaced modes, MIMO technology is the only choice. Test procedures are in general easier. No hammers, No shakers. Modal parameters can be obtained in the serviceability state: Needed for slightly non-linear structures Needed if e.g. operating (rotating) machinery is present. In case of larger structures, traditional modal testing is simply impossible. No hammers or shakers can excite the structure due to the mass and the low frequency modes. Possible applications: FEM updating / validation Damage detection Structural Health Monitoring Vibration level estimation Fatigue estimation.

18. OMAC 2008: 18 Case Study – Launch Vehicle – FEM Validation Launch Vehicle Control System Verification Control system is based on FE model During launch vibration data is acqiured and transmitted to the ground. Based on OMA analysis the FE model is verified due the different states of the launch. Customer: Boeing Integrated Defense Systems, Delta IV, CA, USA

19. OMAC 2008: 19 Case Study – On Road Testing Customer: Mazda, Hiroshima, Japan

20. OMAC 2008: 20 Case Study – Qutub Minar, New Delhi, India Customer: Mazda, Hiroshima, Japan

21. OMAC 2008: 21 OMAC 2008 Schedule 09:00 – 10:40 Technical Session 1 – Introduction to OMA What is Modal Analysis Traditional Modal Analysis versus Operational Modal Analysis Applications of OMA – Some Case Studies 11:10 - 13:00 Technical Session 2 – Analysis of a Real Structure Preliminary (Automatic) Modal Analysis The Frequency Domain Decomposition (FDD) Technique The Stochastic Subspace Identification Technique. Keynote 14:15 – 15:15 Technical Session 3 – Testing in Practice Generating Test Geometry and Plan the Test Testing using Reference Sensors and Moving Sensors 15:35 – 16:30 Technical Session 4 – Rounding Off Last Questions and Answers

22. OMAC 2008: 22 Frequency Domain Decomposition – Step by Step Frequency Domain Decomposition (FDD) Theory Practise Automation Enhanced Frequency Domain Decomposition (EFDD) Theory Practise Curve-fit Frequency Domain Decomposition (CFDD) Theory Practise

23. OMAC 2008: 23 Frequency Domain Decomposition - Theory Showing that  y  the dynamic deflection is a linear combination of the Mode Shapes, the coefficients being the Modal Coordinates.  y(t)  =  1 q 1 (t) +  2 q 2 (t) +  3 q 3 (t) +    +  n q n (t) = ++ +    +

24. OMAC 2008: 24 Frequency Domain Decomposition - Theory Linear system response: Covariance function of system response: Spectral density function obtained by Fourier Transformation: Spectrum is decouple and described by superposition of Single-Degree-Of-Freedom models G(f)G(f) f

25. OMAC 2008: 25 Frequency Domain Decomposition - Theory Theory says... Perform an SVD Extract Mode Shape

26. OMAC 2008: 26 Frequency Domain Decomposition - Theory G(f)G(f) f In the vicinity of the resonance peak of a well-seperated mode:... the first singular vector is a good approximation to the mode shape!

27. OMAC 2008: 27 Frequency Domain Decomposition - Practice Modes are found by picking the peaks of the modes at 1st singular value. Mode shapes estimate is given by the associate singular vector.

28. OMAC 2008: 28 Frequency Domain Decomposition - Practice 2 is a good place for estimating shape  1 from singular vector v 1 3 is a good place for estimating shape  2 from singular vector v 1 s1s1 s2s2 3 v1s1v1s1 v2s2v2s2 a11a11 a22a22 v1s1v1s1 v2s2v2s2 a11a11 a22a22 v2s2v2s2 v1s1v1s1 a22a22 a11a11 2 1 1 23 11 11 11 22 22 22 In case of non-orthogonal mode shapes:

29. OMAC 2008: 29 Frequency Domain Decomposition - Automation Modal Coherence Discriminator function: Low modal coherence: Noise High modal coherence: Modal dominance 1 0.8 0

30. OMAC 2008: 30 Frequency Domain Decomposition - Automation The measured responses of a structure are approximate Gaussian distributed if just a few independent broad-banded random inputs excites the structure. The measured response will only be approximately Gaussian in case of a large number of different harmonic exciation sources. Detection Procedure: 1. Bandpass filter. 2. Normalize to zero-mean and unit variance. 3. Calculate Kurtosis: 4. If γ is significantly different from 3 then most likely not Gaussian. Hamonic Indicators Gaussian Probability Density Function Sinusoidal Probability Density Function

31. OMAC 2008: 31 Frequency Domain Decomposition - Automation Modal Domain Mode property – defined for all modes Defines the frequency region dominated by the mode Definition: The frequency range around a peak where: Modal coherence is higher than a certain threshold, say 0.8, No harmonics present. Damping resonably low.

32. OMAC 2008: 32 Frequency Domain Decomposition - Automation Procedure: Define a search set as all 1st. Singular values from DC to Nyquist. Identify the highest peak in the current search set. Check if peak could be physical (High modal coherence, no harmonic, low damping). If so, establish a modal domain. If not, establish a noise domain. Exclude modal/noise domain from search set. Continue until: 1. Search set is empty. 2. Peak is below defined noise floor. 3. A specified number of modes are found.

33. OMAC 2008: 33 Enhanced Frequency Domain Decomposition - Theory IFFT performed to calculate Correlation Function of SVD function Frequency and Damping estimated from Correlation Function Mode shape is obtained from weighted sum of singular vectors s1s1 s2s2 00 ii MAC = Improved shape estimation from weighted sum: Select MAC rejection level (default 0,8):

34. OMAC 2008: 34 Enhanced Frequency Domain Decomposition - Practice

35. OMAC 2008: 35 Curve-fit Frequency Domain Decomposition - Theory s1s1 s2s2 00 ii Algorithm: 1. Estimate SDOF spectrum G(f). 2. Calculate half-power spectrum P(f) of G(f) 3. Construct the following matrices: 4. Solve the following regression problem: 5. Estimates are then given by: 6. Frequency and damping are obtain from the roots of A(f) Perform SDOF curve-fitting on estimated SDOF model in frequency domain to estimate Frequency and Damping.

36. OMAC 2008: 36 Curve-fit Frequency Domain Decomposition - Practice Perform SDOF curve-fitting on estimated SDOF model in frequency domain to estimate Frequency and Damping. Mode shape is obtained from weighted sum of singular vectors s1s1 s2s2 00 ii

37. OMAC 2008: 37 Frequency Domain Decomposition - Conclusions Frequency Domain Decomposition: Simple peak picking technique that quickly estimates modes even in case of hundreds of measurement channels. Mode shapes are estimated by removing influence of other modes by utilization of the Singular Value Decomposition. Can easially be automated. Enhanced Frequency Domain Decomposition: Frequency and damping determined on the basis of identification of the SDOF model of the mode. The SDOF model is estimated in frequency domain by utilizing the Modal Assurance Criterion. Frequency and damping is estimated from the time domian equivalent SDOF model. Curve-fit Frequency Domain Decomposition: Curve-fit frequency and damping directly in frequency domain.

38. OMAC 2008: 38 SSI procedure Generate compressed input format –Select total number of modes (structural, harmonics, noise) based on apriori knowledge –Select Identification Class »Unweighted Principal Components (UPC); Principal Components (PC); Canonical Variate Analysis (CVA) Estimate Parameters from Stabilization diagram –Select interval of model order candidates (use SVD diagram) –Estimate models(adjust tolerance criteria) –Select the optimal model (use validation) Select and link modes across data sets Stochastic Subspace Identification (SSI) Classes of Identification Data Driven: Use of raw time data Covariance Driven: Use of Correlation functions

39. OMAC 2008: 39 Combined System Model used in SSI Measured Responses Stationary zero mean Gaussian White Noise Excitation Filter (linear, time-invariant) Structural System (linear, time-invariant) Unknown excitation forces Combined System A State Equation Observation (Output) equation Model of the dynamics of the system Model of the output of the system w t : Process noise - v t : Measurement noise - Model order: Dimension of A Discrete-time Stochastic State Space Model

40. OMAC 2008: 40 Stochastic Subspace Identification (SSI) w t :Process noise v t :Measurement noise Modal decomposition Eigenvalues Modal frequency and damping Left hand mode shapes Physical Modes Right hand mode shapes Non-physical Modes Modal distribution of e Initial modal amplitudes Discrete-time Stochastic State Space Model Innovation form e t : Innovation (white noise) K : Kalman gain (noise model) Modal parameter extraction from SSI

41. OMAC 2008: 41 Conclusion: If can be determined then A and C can be optimally predicted using least squares estimation Least squares estimation of A and C Stochastic Subspace Identification (SSI) x x x x x x Assuming properties: Zero mean Gaussian stochastic process Modeled by a state space formulation Then a least squares estimation gives Gaussian white noise residuals Result : x x x x x x Error

42. OMAC 2008: 42 Estimation of state vectors: Stochastic Subspace Identification (SSI) S1S1 O: Compressed input format matrix W 1, W 2 : Weighting matrices S 1 : Subspace matrix Selected subspace Singular value State space dimension s1s1 s2s2 s3s3 s4s4 s5s5 s6s6 is calculated from SVD:

43. OMAC 2008: 43 Stochastic Subspace Identification (SSI) Parametrical Modal estimation requiring apriory knowledge of Model Order Physical Modes as well as Non-physical Modes are estimated How can we separate Physical Modes from Non-physical Modes? Physical modes are repeated for multiple Model orders! Stabilization Diagram Frequency Number of modes in the model 4 5 6 7 + + + + X XX X Stable Modes X X X X Remaining modes are considered as unstable X Estimated parameters not fulfilling apriori knowledge of damping + Stable modes are repeated in two consecutive models fulfilling user defined criteria X X X XX Stable Modes not fulfilling Damping apriori knowledge X X X X + X

44. OMAC 2008: 44 Selection of State Space Dimension Error diagram Model vs. measurements Stabilization diagram

45. OMAC 2008: 45 Selecting proper model order for SSI Final Prediction Error –Fitting error decreases with increasing model order –Parameter uncertainty increases with increasing model order Final Prediction Error Model Order Fitting Error Parameter Uncertainty Optimum Choice

46. OMAC 2008: 46 Effects of time varying systems, 2 Run up/down tests gives good modal parameters, but might have bad SSI validation Loading SystemStructural SystemInput StochasticDeterministic Output

47. OMAC 2008: 47 OMAC 2008 Schedule 09:00 – 10:40 Technical Session 1 – Introduction to OMA What is Modal Analysis Traditional Modal Analysis versus Operational Modal Analysis Applications of OMA – Some Case Studies 11:10 - 13:00 Technical Session 2 – Analysis of a Real Structure Preliminary (Automatic) Modal Analysis The Frequency Domain Decomposition (FDD) Technique The Stochastic Subspace Identification Technique. Keynote 14:15 – 15:15 Technical Session 3 – Testing in Practice Generating Test Geometry and Plan the Test Testing using Reference Sensors and Moving Sensors 15:35 – 16:30 Technical Session 4 – Rounding Off Last Questions and Answers

48. OMAC 2008: 48 The ARTeMIS Software Solution TEAC LX ARTeMIS Testor ARTeMIS Extractor Vibration Data File Transfer Or OLE ARTeMIS Solution

49. OMAC 2008: 49 Data Acquisition – Total Sample Time and Sampling Frequency T Sampling Interval, s t total Total measurement time, s f s Sampling Frequency, Hz f ν Nyquist Frequency, Hz f min Frequency of lowest mode of interest f max Frequency of highest mode of interest

50. OMAC 2008: 50 Acquire High Quality Data Check / Optimize effective dynamic range: –Can be increased by oversampling / decimation. –Check valleys of spectra and SVD.