Experimentele Modale Analyse LES 2 – THEORETISCHE ASPECTEN Patrick Guillaume E-mail: patrick.guillaume@vub.ac.be Tel.: 02/6293566 12-1-2019 EXPERIMENTELE MODALE ANALYSIS, LES 2

Frequency Response of MDOF System EXPERIMENTELE MODALE ANALYSIS, LES 2

Poles of SDOF System Poles : Generalized eigenvalue problem EXPERIMENTELE MODALE ANALYSIS, LES 2

The SDOF Modal-Parameter Model Pole p Re - Decay rate Im - Modal frequency Residue R Strength of the mode EXPERIMENTELE MODALE ANALYSIS, LES 2

Poles and Residues Amplitude of mode is not given by the residue alone Hi-Fi turntable Car Same pole Different residues EXPERIMENTELE MODALE ANALYSIS, LES 2

Multiple Degree of Freedom (MDOF) Dynamic stiffness matrix Transfer function matrix EXPERIMENTELE MODALE ANALYSIS, LES 2

FREQUENCY RESPONSE FUNCTION IMPULSE RESPONSE FUNCTION Poles of MDOF System Poles: FREQUENCY RESPONSE FUNCTION IMPULSE RESPONSE FUNCTION EXPERIMENTELE MODALE ANALYSIS, LES 2

Eigenvalues and Eigenvectors No Damping EXPERIMENTELE MODALE ANALYSIS, LES 2

Weighted Orthogonality of Modal Vectors Pre-multiply with Transpose the equation EXPERIMENTELE MODALE ANALYSIS, LES 2

Weighted Orthogonality of Modal Vectors Substracting From eq. 1: (eq. 1) EXPERIMENTELE MODALE ANALYSIS, LES 2

Modal Mass and Stiffness No Damping Modal mass and stiffness are not unique ! EXPERIMENTELE MODALE ANALYSIS, LES 2

Modal Coordinates EXPERIMENTELE MODALE ANALYSIS, LES 2

Modal Model EXPERIMENTELE MODALE ANALYSIS, LES 2

Modal Decomposition IRF FRF EXPERIMENTELE MODALE ANALYSIS, LES 2

Scaling of the Mode Shapes EXPERIMENTELE MODALE ANALYSIS, LES 2

Proportional Damping Eigenvalues EXPERIMENTELE MODALE ANALYSIS, LES 2

Proportional Damping Complex system poles and normal modal vectors Modal mass, stiffness and damping EXPERIMENTELE MODALE ANALYSIS, LES 2

General Viscous Damping EXPERIMENTELE MODALE ANALYSIS, LES 2

General Viscous Damping Eigenvalues and vectors Complex system poles and complex modal vectors Orthogonality EXPERIMENTELE MODALE ANALYSIS, LES 2

General Viscous Damping Poles: FREQUENCY RESPONSE FUNCTION IMPULSE RESPONSE FUNCTION EXPERIMENTELE MODALE ANALYSIS, LES 2

General Viscous Damping EXPERIMENTELE MODALE ANALYSIS, LES 2

Operational Deflection Shapes (ODS) Resonant frequencies (peaks) Mode shapes (ODS) EXPERIMENTELE MODALE ANALYSIS, LES 2

Operational Deflection Shapes (ODS) Uncoupled modes (SDOF) EXPERIMENTELE MODALE ANALYSIS, LES 2

Operational Deflection Shapes (ODS) Coupled modes + = EXPERIMENTELE MODALE ANALYSIS, LES 2

Force Appropriation Method Also called: Normal Mode Testing Phase Resonance Testing Traditionally used for Ground Vibration Testing (GVT) of airplanes (large structures) First method to use multiple inputs Find (mono-phased) forced vector to obtain a (mono-phased) response vector with a 90 degree phase EXPERIMENTELE MODALE ANALYSIS, LES 2

Force Appropriation Method Real/imaginary part Find (mono-phased) forced vector to obtain a (mono-phased) response vector with a 90 degree phase Verification of normal mode tuning by turning off the excitations EXPERIMENTELE MODALE ANALYSIS, LES 2

MatLab Oefening modal_model.m SDOF MDOF Eigenwaarden en vectoren Schaling van de eigenvectoren EXPERIMENTELE MODALE ANALYSIS, LES 2