Experimentele Modale Analyse

1
Experimentele Modale Analyse

• LES 3 NIETPARAMETRISCHE EN PARAMETRISCHE
  SCHATTINGEN

Patrick Guillaume E-mail patrick.guillaume_at_vub.
ac.be Tel. [phone]
2
Statistic properties of estimators

• Consistency
• Efficiency
• Cramer-Rao Lower Bound

OK NOT OK
OK NOT OK
3
What is Curve Fitting?

• Least-Squares Fit (Static)

4
SISO Errors-in-Variables Model
output
input
5
Noise in the Output Measurement

• Force measurement
• Electrical noise
• Response measurement
• Electrical noise
• Machines, footsteps, wind, sound, will result
  in mechanical noise (process noise)
• Least-Squares Estimation
• Minimize the effect of output noise

6
Noise in the Input Measurement

• At its natural frequencies the structure becomes
  very compliant
• Least-Squares Estimation
• Minimize the effect of input noise

7
The Coherence Function

• Degree of linearity
• Smaller than 1 when
• Noise in the measurements
• Nonlinearities

8
Noise in the Input and Output Measurements

• Choice of optimal FRF estimator
• H1
• Under estimation
• H2
• Over estimation
• Hv, Hiv,

9
MIMO Errors-in-Variables Model
10
Classical MIMO FRF estimators

• H1 estimator (Least Squares)
• H2 estimator (Least Squares)
• Hv estimator (Total Least Squares)

11
Errors-in-Variables Approach

• GTLS
• H1 (LS)
• Hv (TLS)

12
Instrumental Variables
13
FRF estimators for periodic signals

• In theory
• Number of problems
• Mechanical noise in the structure
• Electrical noise in the instrumentation
• Averaging

14
Bias error of FRF estimates

• 1
• 2

1
2
15
Bias error of FRF estimates
16
Empirical TF estimate (ETFE)

• Scalar systems
• Multivariable systems with Ni inputs

17
Periodic Signals 2 Inputs
1
18
Periodic Signals Multivariable Systems

• Errors-in-variables model (synchronized meas.)

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

20
Optimal Experimental Design Multisines
21
Parameter Estimation by Curve Fitting

• Gold in ? Gold out

22
Modal Model is Nonlinear-in-the-Parameters
23
Curve-Fitters for Modal Analysis

• SDOF (Dynamic)

24
Linear Least-Squares Solution

• Over-determined set of real-valued equations
  (mgtn)
• Equation error vector
• LS cost function
• Stationary points

25
Example 1 LS Fit of 1/k (Static)
26
Example 2 LS Fit of SDOF Model
27
Local and Global Curve Fitters

• Poles are global parameters
• Residues are local parameters
• Two step approach

28
Least Squares Complex Exponential LSCE
29
Stabilization Diagram
30
Least Squares Frequency Domain LSFD
31
PZL Mielec Skytruck (FLiTE Project)
32
Mode Shapes (3.17 Hz, 1.62 )
33
Mode Shapes (8.39 Hz, 1.93 )