Mechanische trillingen LES 8 MODALE ANALYSE Patrick Guillaume

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Mechanische trillingen LES 8 – MODALE ANALYSE Patrick Guillaume E-mail: patrick. guillaume@vub. ac. be

Mechanische trillingen LES 8 – MODALE ANALYSE 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 11/30/2020 MECHANISCHE TRILLINGEN, LES 8, 2005

Signal vs. System Analysis Signal Analysis / Fourier Analysis – Only the responses (due

Signal vs. System Analysis Signal Analysis / Fourier Analysis – Only the responses (due to the unknown operating forces) are measured System Analysis / Modal Analysis – The system is stimulated with known forces (in laboratory conditions) – Applied forces and resulting responses are measured 2 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Basic Equations for a 2 -DOF System Forces acting on mass 1 f 2(t)

Basic Equations for a 2 -DOF System Forces acting on mass 1 f 2(t) – m 2 f 1(t) x 2(t) c 2 k 2 m 1 k 1 – x 1(t) c 1 – – 3 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Basic Equations for a 2 -DOF System Time domain Frequency domain (Laplace) Matrix notations

Basic Equations for a 2 -DOF System Time domain Frequency domain (Laplace) Matrix notations Dynamic stiffness matrix Transfer function matrix 4 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

FRF of 2 -DOF System f 2(t) m 2 f 1(t) x 2(t) c

FRF of 2 -DOF System f 2(t) m 2 f 1(t) x 2(t) c 2 k 2 m 1 k 1 x 1(t) c 1 0° -180° -360° 5 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Multiple Degree of Freedom (MDOF) Dynamic stiffness matrix Transfer function matrix 6 MECHANISCHE TRILLINGEN,

Multiple Degree of Freedom (MDOF) Dynamic stiffness matrix Transfer function matrix 6 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

FRF and IRF of MDOF Systems POLES: FREQUENCY RESPONSE FUNCTION (GLOBAL PARAMETERS) IMPULSE RESPONSE

FRF and IRF of MDOF Systems POLES: FREQUENCY RESPONSE FUNCTION (GLOBAL PARAMETERS) IMPULSE RESPONSE FUNCTION 7 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Decomposition IRF FRF 8 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research

Modal Decomposition IRF FRF 8 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Eigenvalues and Eigenvectors No Damping 9 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration

Eigenvalues and Eigenvectors No Damping 9 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

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

Modal Mass and Stiffness No Damping – Modal mass and stiffness are not unique ! 10 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Coordinates 11 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije

Modal Coordinates 11 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Model 12 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije

Modal Model 12 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Scaling of the Mode Shapes 13 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration

Scaling of the Mode Shapes 13 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Model – Modal Parameters Modal participation factors 14 MECHANISCHE TRILLINGEN, LES 8, 2005

Modal Model – Modal Parameters Modal participation factors 14 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Model – Modal Parameters 15 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration

Modal Model – Modal Parameters 15 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Model – Modal Parameters 16 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration

Modal Model – Modal Parameters 16 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Mode Shape Vector Mode shape is a deflectionpattern associated with a modal frequency –

Mode Shape Vector Mode shape is a deflectionpattern associated with a modal frequency – or pole Relative displacements Mode shapes are continuous functions of position Sampled mode shapes = mode shape vector “Spatial resolution” and “Spatial aliasing” 17 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Normal and Complex Modes Normal modes (or real modes) – All points are moving

Normal and Complex Modes Normal modes (or real modes) – All points are moving in phase or out of phase – Node points Complex modes – Mode shape vectors are complex – Node points are not well defined anymore Warning: Poor measurements 18 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Cantilever Beam 19 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije

Cantilever Beam 19 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

FRF in respectievelijk knooppunt 1 en 2 Mode 1 2 3 20 MECHANISCHE TRILLINGEN,

FRF in respectievelijk knooppunt 1 en 2 Mode 1 2 3 20 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Real-Imaginary Plot – Peak Picking (normal modes) 21 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics

Real-Imaginary Plot – Peak Picking (normal modes) 21 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Simple Structure – Mode Shapes 22 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration

Simple Structure – Mode Shapes 22 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Peak Picking 23 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije

Peak Picking 23 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Shaker Testing - Fixed FRF Measurements 24 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics &

Shaker Testing - Fixed FRF Measurements 24 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Impact Testing - Roving FRF Measurements 25 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics &

Impact Testing - Roving FRF Measurements 25 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Roving or Fixed FRF Measurements 26 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration

Roving or Fixed FRF Measurements 26 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Noise in the Output Measurement Force measurement – Electrical noise Response measurement – Electrical

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 27 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Noise in the Output Measurement – Multivariable 28 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics

Noise in the Output Measurement – Multivariable 28 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Noise in the Input Measurement At its natural frequencies the structure becomes very compliant

Noise in the Input Measurement At its natural frequencies the structure becomes very compliant Least-Squares Estimation – Minimize the effect of input noise 29 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

The Coherence Function Degree of linearity Smaller than 1 when … – Noise in

The Coherence Function Degree of linearity Smaller than 1 when … – Noise in the measurements – Nonlinearities 30 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Noise in the Input and Output Measurements Choice of optimal FRF estimator – H

Noise in the Input and Output Measurements Choice of optimal FRF estimator – H 1 – Under estimation – H 2 – Over estimation – H 3, Hv, Hiv, … 31 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Implementing the Excitation Attached exciters – – Electrodynamic exciter Hydraulic exciter Eccentric rotating masses

Implementing the Excitation Attached exciters – – Electrodynamic exciter Hydraulic exciter Eccentric rotating masses More exotic devices such as rockets, … Non-attached exciters – Hammers, impactors – Acoustic excitation 32 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Force Measurements Piezo-electric force transducer – – Small size and mass Linearity Wide dynamic

Force Measurements Piezo-electric force transducer – – Small size and mass Linearity Wide dynamic range Wide frequency band Exciter attachment – “Stringer” – High axial stiffness – Low transverse and rotational stiffness – Mechanical “fuse” – Protection against destructive overload 33 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Random Excitation Random sequence with Gaussian distribution – Random amplitudes and phases Averaging is

Random Excitation Random sequence with Gaussian distribution – Random amplitudes and phases Averaging is needed – Converge to flat amplitude spectrum – Remark: Force is not flat in general due to interaction with the structure Signal processing errors (leakage errors) Effect of nonlinearities is reduced by averaging 34 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Random Noise 35 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije

Random Noise 35 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Uniform Window 36 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije

Uniform Window 36 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Hanning Window 37 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije

Hanning Window 37 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Burst Random 38 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije

Burst Random 38 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Burst Random with Uniform Window 39 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration

Burst Random with Uniform Window 39 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Exponential Window 40 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije

Exponential Window 40 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Periodic Random / Pseudo-Random Excitation Periodic random – No leakage errors (periodic signal) in

Periodic Random / Pseudo-Random Excitation Periodic random – No leakage errors (periodic signal) in steady-state conditions – Averaging is required Pseudo random – Constant amplitudes and random phases – No leakage errors (periodic signal) in steady-state conditions – Averaging is not required 41 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Impact Excitation Force transducer and tip Advantages – Easy to use – No interaction

Impact Excitation Force transducer and tip Advantages – Easy to use – No interaction with structure – Force is flat in useful frequency range – Relatively inexpensive 42 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Impact Excitation Disadvantages – Large crest factor – Nonlinearities – Limited control of amplitude

Impact Excitation Disadvantages – Large crest factor – Nonlinearities – Limited control of amplitude spectrum 43 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Force Window Force window (or transient window) – Remove noise Effect of “Double Hits”

Force Window Force window (or transient window) – Remove noise Effect of “Double Hits” 44 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Response Window Lightly damped structure – Leakage errors Heavily damped structure – Remove noise

Response Window Lightly damped structure – Leakage errors Heavily damped structure – Remove noise 45 MECHANISCHE TRILLINGEN, LES 8, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel