Content A Polyreference Least Square Complex Frequency Introduction - PowerPoint PPT Presentation

Content A Polyreference Least Square Complex Frequency Introduction domain based statistical test for damage detection Frequency domain modal analysis Gilles Canales, Laurent Mevel, Mich` ele Basseville Scalar frequency domain local test for change detection IRISA (U. Rennes 1 & INRIA & CNRS), Rennes, France Multidimensional frequency domain local test Eurˆ eka project no 3341 FliTE2 Model validation Application example michele.basseville@irisa.fr -- http://www.irisa.fr/sisthem/ Conclusion 1 2 Introduction Modal model and parameters • Damage detection � � − 1 M s 2 + C s + K M , C , K ∈ R N m × N m Y( s ) = H( s )F( s ) , H( s ) = , • Local approach to change detection ← → parameter estimating function Full modal model: λ m : m -th mode, Φ m ∈ C N m associated modeshape • Time domain: subspace-based χ 2 -tests, � � input-output or output-only N m Φ ∗ m Φ ∗ T � Φ m Φ T + Q ∗ m m H( s ) = Q m m s − λ ∗ s − λ m m • Limitation: number of outputs m =1 Limited modal model: N i inputs, N o outputs, N i ≪ N o • Frequency domain: new input-output identification algorithm � � (Polyreference LSCF) N m + Φ ∗ m L ∗ T � Φ m L T m m H( s ) = s − λ ∗ s − λ m • Wanted: associated damage detection test m m =1 L m ∈ C N i : modal participation factors Nominal input/output transfer functions available 3 4

Common denominator transfer function model Scalar frequency domain test for change detection Local approach to testing G = G 0 for the input/output transfer Ω l = e iω l T s , Polynomial basis function (Ω l ) 1 ≤ l ≤ N f , T s sampling function (Benveniste-Delyon, 2000) 1 Common denominator transfer function model � G − G 0 = √ y n = G ( z ) u n + v n , G K H(Ω l ) = B(Ω l ) A − 1 (Ω l ) , B , A polynomials DFT on K blocks with size N : ( U N k ( ω )) k =1 ...K , ( Y N k ( ω )) k =1 ...K FRF between all the inputs and any output o √ K K, N → ∞ , → 0 H o (Ω l ) = B o (Ω l ) A − 1 (Ω l ) N � � Modal analysis algorithm (Guillaume, 2006) � K ∆ ζ N 1 k =1 U N Y N k ( ω ) − G 0 (Ω) U N √ k ( − ω ) K ( G 0 , ω ) = k ( ω ) � � K � Measured FRFs H o ( ω l ) � � o,k S uu ( ω ) � G (Ω) , S uu ( ω ) S vv ( ω ) ∼ N Minimize the LS cost function � � � � E H E o ( ω l ) = � H o ( ω l ) A(Ω l ) − B o (Ω l ) C = trace o ( ω l ) E o ( ω l ) , | ζ N K ( G 0 , ω ) | 2 G (Ω) � = 0 : χ N Test � G (Ω) = 0 / � K ( G 0 , ω ) = o l S uu � 0 ( ω ) � S vv 0 ( ω ) 5 6 Multidimensional frequency domain local test Use numerator and denominator of common-denominator TF Model validation 1 ˜ H − H 0 = √ B( ω ) = H( ω ) A( ω ) + V( ω ) , H K • One data set Reference FRFs → on K N -size blocks: ( A N 0 ,k (Ω) , B N 0 ,k (Ω)) k =1 ,... ,K B N k, 0 (Ω) = H 0 ( ω ) A N k, 0 (Ω) + V N • Does it match the reference modal model ? k, 0 ( ω ) Does it match slight modifications of the modal model ? New FRFs (H( ω ℓ )) ℓ =1 ,... ,N f . For each ω = ω ℓ : � � � � H � K ∆ 1 ζ N K (B N k, 0 , A N B N k, 0 (Ω) − H( ω ) A N A N k, 0 , ω ) = √ k, 0 (Ω) k, 0 (Ω) → Optimizing the χ 2 -test criterion • − k =1 K � � − ˜ H( ω ) S aa 0 ( ω ) , S aa 0 ( ω ) S vv ∼ N 0 ( ω ) √ • Implementation: Rule of thumb K ∼ N k, 0 , ω ) = ζ N K (B , A , ω ) ( ζ N K (B , A , ω )) H ( ω ) � =0 : χ N K (B N k, 0 , A N Test � H ( ω )=0 / � � 0 ( ω ) � S aa S vv 0 ( ω ) 7 8

Example - Aircraft in-flight test data Example - Numerical results (Cauberghe PhD, 2004) • N i = 1 , N o = 7 - Artificial excitation • Reference : Modal analysis on temporal data set, n = 24000 First mode : 98 . 7 Hz χ 2 -test, entire frequency band χ 2 -test, at the 1st frequency Second mode : 201 . 3 Hz Varying perturbation on mode 1 Section along perturbation axis 275 . 7 Hz Third mode : → B N k, 0 , A N • K = 28 blocks with size N = 784 − k, 0 • 1st mode changed from 95% to 105% FRFs re-built under every change condition Entire band, -1 % perturbation Entire band, -3 % perturbation 9 10 Conclusion Frequency domain test for change detection Polyreference LSCF Local approach to change detection Multidimensional test Relevance for model validation on a real aircraft Ongoing and future issues: Output-only detection algorithm (OMAX) Damage localization Large number of outputs 11