Experimentele Modale Analyse LES 2 THEORETISCHE ASPECTEN Patrick

Frequency Response of MDOF System 2 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration

Poles of SDOF System l Poles : l l l Generalized eigenvalue problem 3

The SDOF Modal-Parameter Model Pole p – Re - Decay rate – Im -

Poles and Residues Amplitude of mode is not given by the residue alone Hi-Fi

Multiple Degree of Freedom (MDOF) Dynamic stiffness matrix Transfer function matrix 6 EXPERIMENTELE MODALE

Poles of MDOF System Poles: FREQUENCY RESPONSE FUNCTION IMPULSE RESPONSE FUNCTION 7 EXPERIMENTELE MODALE

Eigenvalues and Eigenvectors No Damping 8 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration

Weighted Orthogonality of Modal Vectors – Pre-multiply with – Transpose the equation 9 EXPERIMENTELE

Weighted Orthogonality of Modal Vectors (eq. 1) – – Substracting – – From eq.

Modal Mass and Stiffness No Damping – Modal mass and stiffness are not unique

Modal Coordinates 12 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije

Modal Model 13 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije

Modal Decomposition IRF FRF 14 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research

Scaling of the Mode Shapes 15 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration

Proportional Damping – Eigenvalues 16 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research

Proportional Damping – Complex system poles and normal modal vectors – Modal mass, stiffness

General Viscous Damping 18 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group

General Viscous Damping – Eigenvalues and vectors – Complex system poles and complex modal

General Viscous Damping Poles: FREQUENCY RESPONSE FUNCTION IMPULSE RESPONSE FUNCTION 20 EXPERIMENTELE MODALE ANALYSIS,

General Viscous Damping 21 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group

Operational Deflection Shapes (ODS) Resonant frequencies (peaks) Mode shapes (ODS) 22 EXPERIMENTELE MODALE ANALYSIS,

Operational Deflection Shapes (ODS) Uncoupled modes (SDOF) 23 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics

Operational Deflection Shapes (ODS) Coupled modes + = 24 EXPERIMENTELE MODALE ANALYSIS, LES 2

Force Appropriation Method – Also called: – Normal Mode Testing – Phase Resonance Testing

Force Appropriation Method – Real/imaginary part – Find (mono-phased) forced vector to obtain a

Mat. Lab Oefening modal_model. m – – SDOF MDOF Eigenwaarden en vectoren Schaling van

Slides: 27

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Experimentele Modale Analyse LES 2 – THEORETISCHE ASPECTEN Patrick Guillaume E-mail: patrick. guillaume@vub. ac. be Tel. : 02/6293566 Faculty of Engineering Department of Mechanical Engineering ACOUSTICS & VIBRATION RESEARCH GROUP Pleinlaan 2 • B-1050 Brussel • Belgium avrg@vub. ac. be • http: //avrg. vub. ac. be 1 6/5/2021 EXPERIMENTELE MODALE ANALYSIS, LES 2

Frequency Response of MDOF System 2 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Poles of SDOF System l Poles : l l l Generalized eigenvalue problem 3 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

The SDOF Modal-Parameter Model Pole p – Re - Decay rate – Im - Modal frequency Residue R – Strength of the mode 4 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Poles and Residues Amplitude of mode is not given by the residue alone Hi-Fi turntable Car – Same pole – Different residues 5 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Multiple Degree of Freedom (MDOF) Dynamic stiffness matrix Transfer function matrix 6 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Poles of MDOF System Poles: FREQUENCY RESPONSE FUNCTION IMPULSE RESPONSE FUNCTION 7 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Eigenvalues and Eigenvectors No Damping 8 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Weighted Orthogonality of Modal Vectors – Pre-multiply with – Transpose the equation 9 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Weighted Orthogonality of Modal Vectors (eq. 1) – – Substracting – – From eq. 1: 10 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Mass and Stiffness No Damping – Modal mass and stiffness are not unique ! 11 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Coordinates 12 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Model 13 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Decomposition IRF FRF 14 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Scaling of the Mode Shapes 15 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Proportional Damping – Eigenvalues 16 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Proportional Damping – Complex system poles and normal modal vectors – Modal mass, stiffness and damping 17 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

General Viscous Damping 18 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

General Viscous Damping – Eigenvalues and vectors – Complex system poles and complex modal vectors – Orthogonality 19 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

General Viscous Damping Poles: FREQUENCY RESPONSE FUNCTION IMPULSE RESPONSE FUNCTION 20 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

General Viscous Damping 21 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Operational Deflection Shapes (ODS) Resonant frequencies (peaks) Mode shapes (ODS) 22 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Operational Deflection Shapes (ODS) Uncoupled modes (SDOF) 23 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Operational Deflection Shapes (ODS) Coupled modes + = 24 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

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 25 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

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 26 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Mat. Lab Oefening modal_model. m – – SDOF MDOF Eigenwaarden en vectoren Schaling van de eigenvectoren 27 EXPERIMENTELE MODALE ANALYSIS, LES 2 Acoustics & Vibration Research Group Vrije Universiteit Brussel