1. Experimentele Modale Analyse
LES 3 – NIETPARAMETRISCHE EN PARAMETRISCHE SCHATTINGEN
Patrick Guillaume
Tel.: 02/
1/13/2019
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

2. Statistic properties of estimators
Consistency
Efficiency
Cramer-Rao Lower Bound
OK NOT OK
OK NOT OK
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

3. What is Curve Fitting? Least-Squares Fit (Static)
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

4. SISO “Errors-in-Variables” Model
input
output
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

5. Noise in the Output Measurement
Force measurement
Electrical noise
Response measurement
Machines, footsteps, wind, sound, … will result in mechanical noise (process noise)
Least-Squares Estimation
Minimize the effect of output noise
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

6. Noise in the Input Measurement
At its natural frequencies the structure becomes very compliant
Least-Squares Estimation
Minimize the effect of input noise
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

7. The Coherence Function
Degree of linearity
Smaller than 1 when …
Noise in the measurements
Nonlinearities
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

8. Noise in the Input and Output Measurements
Choice of optimal FRF estimator H1
Under estimation H2
Over estimation
Hv, Hiv, …
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

9. MIMO “Errors-in-Variables” Model
input 1
input 2
input Ni
output 1
output 2
output No
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

10. Classical MIMO FRF estimators
H1 estimator (Least Squares)
H2 estimator (Least Squares)
Hv estimator (Total Least Squares)
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

11. “Errors-in-Variables” Approach
GTLS
H1 (LS)
Hv (TLS)
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

12. Instrumental Variables
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

13. FRF estimators for periodic signals
In theory
Number of problems
Mechanical noise in the structure
Electrical noise in the instrumentation
Averaging
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

14. Bias error of FRF estimates1 2 1 2
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

15. Bias error of FRF estimates
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

16. Empirical TF estimate (ETFE)
Scalar systems
Multivariable systems with Ni inputs
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

17. Periodic Signals – 2 Inputs
output 1
output 2
output 3
– 1
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

18. Periodic Signals – Multivariable Systems
Errors-in-variables model (synchronized meas.)
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

19. Optimal Experimental Design
D-optimal design = find the amplitude-constrained inputs that minimizes the determinant of the CRLB (Cramer-Rao Lower Bound)
Stepped-sine excitation
2 inputs: (0°, 0°), (0°, 180°)
3 inputs: (0°, 0°, 0°), (0°, 120°, -120°), (0°, -120°, 120°)
Multisine excitation
Hadamard matrix
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

20. Optimal Experimental Design – Multisines
Hadamard matrix
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

21. Parameter Estimation by Curve Fitting
Gold in  Gold out
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

22. Modal Model is Nonlinear-in-the-Parameters
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

23. Curve-Fitters for Modal Analysis
SDOF (Dynamic)
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

24. Linear Least-Squares Solution
Over-determined set of ‘real-valued’ equations (m>n)
Equation error vector
LS cost function
Stationary points
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

25. Example 1 – LS Fit of 1/k (Static)
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

26. Example 2 – LS Fit of SDOF Model
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

27. Local and Global Curve Fitters
Poles are global parameters
Residues are local parameters
Two step approach
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

28. Least Squares Complex Exponential – LSCE
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

29. Stabilization Diagram= =
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

30. Least Squares Frequency Domain – LSFD
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

31. PZL Mielec Skytruck (FLiTE Project)
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

32. Mode Shapes (3.17 Hz, 1.62 %)
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

33. Mode Shapes (8.39 Hz, 1.93 %)
EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005