DEVELOPMENT AND USE OF OPERATIONAL MODAL ANALYSIS

Transcription

1 Università degli Studi di Napoli Federico II DEVELOPMENT AND USE OF OPERATIONAL MODAL ANALYSIS Examples of Civil, Aeronautical and Acoustical Applications Francesco Marulo Tiziano Polito 27 Ma 2008 Structural and Geotechnical Dnamic Laborator STREGA Engineering Facult - Universit of Molise Campobasso - Ital

2 Overview Introduction Tpical Applications Theoretical Background Numerical assessment Case Studies Discussion of Results Concluding Remarks 2

3 Introduction The operational modal analsis (OMA) is an added tool for the continuing improvement of the man s s products Real time measurements and analsis represents an invaluable process for gaining true experiences on structural dnamic behaviour OMA advantages: Real structures exhibits true dnamic response OMA drawbacks: Hpothesis on the forcing functions No pre-studies Onl retrofit Expert user heav math 3

4 Theoretical Background OMA s objective is the modal parameters identification from output- onl measurements The classical structural equation of motion: [ M ]{&& x() t } + [ C] { x& ( t) } + [ K] { x( t) } = { F( t) } = [ D] { u( t) } ma be written as: where x k = x A = exp ( kδt ) ( A t) c Δ [ ] x k +1 k = Ax = Cx k k + + Bu Du k k + w + v [ 0] [ I ] [ ] [ ] [ ] [ ] 1 M K M C A = c 1 k k 1 [ B] = ([ A] [ I ])[ ] [ ] [ ] [ 0] [ ] [ ] M = B B c 1 A c B c Discrete-time time stochastic state-space space model under the assumption of white noise or time impulse excitation 4

5 a b c d e b c d e f c d e f g d e f g h e f g h i Hankel matrix A Hankel matrix is a square matrix that is smmetric and constant across the anti-diagonals Hankel matrices are formed when given a sequence of output data and a realization of an underling state-space space or hidden Markov model is desired. The Singular Value Decomposition (SVD) of the Hankel matrix provides a means of computing the A,B, and C matrices which define the state-space space realization 5

6 Methods for OMA Data Driven (DD) This algorithm is based on the following definition of the Hankel matrix H = 1 N 0 1 i 1 i i+ 1 2i i i+ 1 i+ 2 2i N 1 N i+ N 2 i+ N 1 i+ N 2i+ N 2 Y = Yi 0 i 1 2i 1 Yp = Y which projects future outputs into the space of the past outputs ts f Again the singular value decomposition provides, through the observabilit and controllabilit matrices, the structural modal parameters 6

7 Methods for OMA (cont d) Covariance Data Driven (CDD) Based on the Hankel matrix built as [ H ] p, q = [ R1 ] [ R2 ] L [ Rq ] [ R ] [ ] [ ] 2 R3 L Rq+ 1 [ ] [ ] [ ] L L L L R R L R p p+ 1 p+ q 1 S k 1 1 [ R ] = { }{ } which, eventuall through a weighting process,, can be realized in the observabilit and controllabilit matrices [ ][ ][ ] [ ] [ ] [ S ] [ ] H p, q W2 = U1 U 2 [] 0 [] 0 Choice of the weighting matrices ma help for an improved sstem identification k S [ V1 ] [ V ] T W T = k s= 0 k+ s s [ U ][ S ][ V ] T T 7

8 Developed in the Matlab environment In-house Software AMOp Text format for the input data files (different sources) Useful for both novice and expert user Starting screen Stabilization diagram Geometr module 8

9 6 dof s sstem Numerical Simulation The numerical simulation appears to be a viable tool for checking the algorithms and the developed software Results Reference Values fn ξn Response to Impulse Excitation Response to Random Excitation 9

10 Numerical Results Reference Values f n ξ n Impulse Exc. f n ξ n Random Exc. f n ξn Impulse Exc. fn ξn Random Exc fn ξn AMOp CDD AMOp DD Robust methodolog 10

11 Simulation of a Real Structure Finite element analsis of a bridge Modal parameters easil computed numericall Simulation with impulse forcing function 6 x 10-8 Grid Point #3 - Acceleration Time Histor 2.5 x 10-7 Grid Point #10 - Acceleration Time Histor Amplitude time [sec] Amplitude time [sec] Close to excitation Far from excitation 11

12 Flight Testing In Flight measurement of the vertical fin & rudder vibration behaviour Objective: To establish variation of the modal parameters with speed Input Force: Pilot induced excitation through pedals mixed with air turbulence 12

13 Example of measured acceleration time-histories Fin Flight Test Results Rudder Input V=180 Km/h V=220 Km/h Rudder & Fin - Test Case A Results 180 km/h mode 1 mode 2 mode 3 mode 4 DD CDD NI NI km/h mode 1 mode 2 mode 3 mode 4 DD NI NI CDD NI NI 220 km/h mode 1 mode 2 mode 3 mode 4 DD NI NI CDD NI NI f [Hz] ξ f [Hz] ξ f [Hz] ξ f [Hz] ξ 13

14 Testing on Civil Structures Bridge deck motorwa Identification of the first structural mode-shapes (model correlation, structural monitoring, ) 1 st Traffic excitation Freq. [Hz] Damp. [%] DD CDD DD CDD 0,89 0,91 0,04 0,04 1,15 1,20 0,03 0,04 2,44 2,03 0,05 0,06 3,01 2,95 0,04 0,04 3,75 3,65 0,06 0,06 Step bumper truck 3 rd 14

15 SMall Acoustic Research Facilit (SMARF) Acoustic Application Facilit for the measurement of the Transmission Loss (TL) of Structural Panels 15

16 Receiving Room Microphone Setup #1 Microphone Setup #2 Run 1 Speaker Run 2 Modal Hammer Run 3 Modal Hammer Run 4 Modal Hammer Run 5 Modal Hammer Run 6 Speaker 16

17 AMOp Application Examples of Stabilization Diagrams 17

18 Num Exp Correlation OMA vs. FEM Comparison Receiving Room 18

19 Soundproofing Effect Time Microphone histor comparison Setup #3 with and without soundproofing treatment Run 7 Speaker (without( soundproofing) Run 8 Speaker (with( soundproofing) 19

20 Damping Behavior OMA computed damping tendenc with and without soundproofing wall treatment 20

21 Conclusions Operational Modal Analsis is an important tool, eventuall the onl one, for the structural identification of real structure Two correlation-driven stochastic subspace techniques have been used on both simulated and real measurements Efficienc of the methodolog even in high modal densit environment Good abilit and coherence in identifing the damping behaviour It ma need an expert user, for a correct interpretation of the input data and identified results More reliable results are generall obtained combining all the available a information on the tested structure, properl weighted 21