Experimentele Modale Analyse

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Presentation transcript:

Experimentele Modale Analyse LES 3 – NIETPARAMETRISCHE EN PARAMETRISCHE SCHATTINGEN Patrick Guillaume E-mail: patrick.guillaume@vub.ac.be Tel.: 02/6293566 1/13/2019 EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

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

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

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

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

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

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

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

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

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

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

Instrumental Variables EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

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

Bias error of FRF estimates 1 2 1 2 EXPERIMENTELE MODALE ANALYSIS, LES 3, 2005

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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