ONLINE MONITORING OF VARYING MODAL PARAMETERS BY OPERATING MODAL ANALYSIS AND MODEL UPDATING

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1 ONLINE MONIOING OF VAYING MODAL PAAMEES BY OPEAING MODAL ANALYSIS AND MODEL UPDAING V.H. Vu, M. homas, A.A. Lakis and L. Marcouiller 3 Department of Mechanical Engineering, École de technologie supérieure, Montréal, Qc, H3C K3, Canada Department of Mechanical Engineering, École Polytechnique, Montréal, Qc, H3C 3A7, Canada 3 Hydro-Québec s research institute, Varennes, Qc, J3X S, Canada ABSAC An algorithm for the online monitoring of varying modal parameters is presented with the ability to update the model order and with the excitation forces unknown. he solution is found by applying a recursive multivariable least squares method via the computation of the Q factorization on a vector autoregressive model. A minimum description length criterion was utilized to select an optimal model order. he model order may be updated either from a minimum order, from a selected maximum order or from previous computational samples. In all these three methods, only submatrices need to be manipulated. Numerical and experimental data are presented for validation. hree monitoring routines combining the three previous methods are explored. It is shown that the natural frequency variation can be easily identified but the identification of damping variation is more difficult depending with the random excitation in a noisy environment. Keywords: Autoregressive model; recursive least squares; Q factorization updating; model order selection; modal parameter identification; varying system.. VECO AUOEGESSIVE MODEL FO MODAL ANALYSIS he identification of structural modal parameters [] plays an important role in structural health monitoring and is conducted using experimental modal analysis methods, usually in a frequency domain in a wide range of applications []. Further, some advanced methods have further been successfully implemented in user-friendly commercial software applications. However, in several industrial applications [3] where it is possible to stop a machine, the forces can very often not be measured or known and so operating modal analysis must therefore be conducted [4, 5]. Since the forces result from natural excitations, operational modal analysis should deal with the time domain ehich means that the monitoring of the structural modal parameters becomes crucial, especially in structural health monitoring. Examples of such an industrial application can be found in bridge monitoring [6, 7], in identification of mass and damping applied to fluid-structure interactions [8, 9], in crack detection [0] and damage monitoring on structures [, ]..

2 he time domain has been found to be more suitable for operational modal analysis [3, 4] and several well known methods can be cited for the identification of a free vibration temporal response data, such as Ibrahim time domain method (ID) [5], the Least squares complex exponential (LSCE) [6], etc. Since the time domain is suitable for the operational modal analysis, we see that time series model based method can be applied to modeling the data. Assuming a random situation, the excitation may be ignored and since the modal analysis required multiple measurement locations, a vector autoregressive model should be preferred [7] with a d sensor dimension and can be expressed as follows: y( t) a y( t ) a y( t )... a y( t p) e( t) Or y( t) A ( t) e( t) d ddp dp d p () () where A a a... ai... a ddp p is the model parameter matrix, a is the matrix of autoregressive parameters relating the output y( t i) to () i d d y t i d ( ) (i=:p) is the output vector with delay time i, yt, is the sampling period (s), ( t ) y ( t ) ; y ( t ) ;...; y ( t p ) dp is the regressor for the output vector yt (), et () d is the residual vector of all output channels considered as the error of model. his formulation assumes that at instant t, each component of the response vector y i (t) at each location (i) depends on the p previous responses y j (t-k) at each other locations (j). If the data are assumed to be measured in a white noise environment, the least squares estimation is applicable. If N successive output vectors of the responses from yt () to y( t N ) are

3 considered (N dp+d), the model parameters matrix A ddp computation of the Q factorization as follows: A ( ).( ) ( ) can be expressed via the (3) In this formula, and are submatrices of the upper triangular factor derived from the Q factorization of the data matrix as follows: K Q. where Q N N is an orthogonal matrix (that is Q. Q I ), dpdp 0 N( dpd ) dxd 0 0 dpd has the form (4) (5) and data matrix K is constructed from N successive samples: K N dpd ( t) dp y( t) ( t ) y( t ) dp ( t N ) y( t N ) dp d d d (6) Once the model parameters matrix has been estimated, modal parameters can be directly identified from the eigendecomposition of the state matrix. i I I a a a... a dd dd dd p dd dpdp I 0 (7) As earlier indicated, finding the optimal model order is crucial in model based methods [8-]. his optimal order can be independently obtained at the global minimum of an order selection criterion such as AIC [] or MDL and other variants [3]. Our previous research has illustrated 3

4 that the MDL is a very good criterion for short data modelling which is suitable for the monitoring. log( et ˆ( ) ) dp. MDL( p) log( log N) d N (8) where et ˆ( ) is a norm of the estimated model error. It was found that in machinery and structural monitoring, we would want to carry out an online survey of changes in modal parameters, which therefore makes the algorithm update process crucial. Since the optimal model order can change, the conventional recursive least squares updating algorithm [4] presents a certain amount of difficulty. In this paper, the parameters matrix is computed via the Q factorization and three methods are presented in order to update the solution with respect to both time and model order where the optimal order is obtained directly from a previous time scheme.. MEHOD : UPDAING IN IME he Q factorization of a model order p should be recursively updated when a new set of samples data is available along with measuring time. hat means we want to calculate, from the ( k s) matrices Q and of data matrix K at time t k, the Q ( k s) and of data matrix ( k s) K K which is found by deleting the first s rows and appending more s rows to matrix Nx( dpd ) ( k) y ( k) ( k ) y ( k ) ( k N ) y ( k N ) K. (9) K ( ks) Nx( dpd ) ( k s) y ( k s) ( k s ) y ( k s ) ( k s N ) y ( k s N ) (0) he relationship can first be established: K ( k) y ( k) ( k N ) y ( k N ) ( k s ) y ( k s ) ( ks) ( k s N ) y ( k s N ) K ( N s) x( dpd ) ( N s) x( dpd ) () 4

5 hat gives in terms of the Q decomposition of the data matrix: ( k) ( k) Q ( k) y ( k) ( k N ) y ( k N ) ( k s ) y ( k s ) ( k s) ( k s) ( k s N ) y ( k s N ) Q ( N s) x( dpd ) ( N s) x( dpd ) () and in innovative form: where ( k) y ( k) Q 0 ( k N ) y ( k N ) Is ( k s) 0 I Q s ( k s ) y ( k s ) ( ks) ( k s N ) y ( k s N ) I is the unity matrix. s sxs (3) In this algorithm, we want to have an update of the submatrices, and of matrix as ( k s) defined in equation (5). Matrices and should therefore be partitioned in a well conditionned form: ( k) ( k) ( k) y ( k) Q 0 ( k N ) y ( k N ) Is ( ks) 0 I Q s ( k s ) y ( k s ) ( k s) ( k s) ( k s N ) y ( k s N ) where the new submatrices and are related to the submatrices, and as follows: Nxdp 0 dpxdp ( N dp) xdp and Nxdp dpxd ( N dp) xd (4) (5) If the first dp columns of equation (4) are extracted, we obtain: ( k) Q 0 ( k N ) Is 0... ( ks) 0 I... 0 Q s ( ks) ( ks) ( k s N ) (6) 5

6 It can be seen that equation (6) is a sub-problem of equation () for the first dp columns. he right hand side can then be transformed from the left one by using two sets of orthogonal Givens rotations. he first set G applies on the matrix ( k N ) to annihilate all dp s lower elements... ( k s N ) and to obtain an upper triangular matrix. ( k) ( k) ( k N ) ( k N ) G [( J N, dp... J N s, dp )...( J N,... J N s,)] ( k s N ) ( k s N ) where i, j ( N s)( N s) previous step th J is the ( i, j ) Givens matrix zeroing the ( i, j ) th element of the matrix in the ( k N ) Ji, j... J N s, j...( J N,... J N s,).... ( k s N ) hese Givens matrices are easily constructed in common form J i, j ( N s)( N s) c... s... 0 j 0... s... c... 0 i j i c s where Givens( element( j, j), element( i, j)) s c of the matrix of the previous step. (7) (8) he left side term of (6) can be rewritten as: ( k) ( k) ( k) Q 0 ( k N ) Q 0 ( k N ) ( k) ( k N ) G G Q G 0 Is... 0 Is ( k s N ) ( k s N ) ( k s N ) (9) 6

7 he second set of Givens rotations G is used to set unitary the first s rows and columns of the augmented matrix Consider vector z Q ( k) Q 0 0 G Is ( k) ( k) r ( N s) x r ( ) q where q k r is the ( r ) th computational step ( r : s), since have ( k) ( k) r r r r N s r. th r row of the augmented matrix Qr ( k) ( k) z r is orthonormal ( z r z ), we can G z J J... J z (0) where the Givens matrix J zeroing the i th element of i r z is in the form: r at J i ( N s)( N s) c s... 0 ( i ) 0... s c... 0 ( i) ( i) ( i) () Computing the matrix G hence shows it to be equal to the multiplication of s components G G () r rs he left side terms of (6) and (9) are thus further rewritten: ( k) ( k) ( k) ( k) Q 0 ( k N ) Q 0 ( k N ) G G GG 0 Is... 0 Is... ( k s N ) ( k s N ) (3) Since the second Givens rotation set is established on the first s rows of the augmented matrix, two interesting consequences are found: Its right transpose multiplication will unitary the first s rows and first s columns of augmented matrix ( k) Q 0 Q G ; 0 Its left multiplication will nonzero the first s elements of each row of the upper triangular matrix G, making each one an upper Hessenberg matrix. ( k N) 7

8 hat explains: Q 0 Is 0 GG *( ks) 0 I 0 Q s ( k) ( k N )... GG... ( ks) *( ks) ( k s N ) It can be seen that two Givens rotations sets are built only on the first dp columns of the data *( k s) matrix. he derived matrix Q ( k s) therefore coincides to the exact matrix Q on the first dp *( ks) *( ks) *( k s) columns and the orthonormal condition Q Q I is assured. Matrix which is ( k s) only nonzero on the first dp rows is the actual desired matrix hence *( ks ) ( ks ). hen ( k s) the factorized matrices at the sample index ( k s) are therefore exactly updated at this stage: dp (6) ( ks) *( ks) (:,:) (4) (5) With the derived matrix Q *( k s), the last d columns of equations (5) can be rewritten as: y ( k) Q 0 y ( k N ) Is 0... *( ks) 0 I... 0 Q s y ( k s ) *( ks) y ( k s N ) *( k ) and the matrix is directly extracted from the equality y ( k) 0 ( )... I k s Q 0 y ( k N ) *( k s) y ( k s ) 0 Q 0 Is... *( ks) y ( k s N ) (7) (8) *( k s) As discussed earlier, the first dp of matrix are also exactly derived, which means the ( k s) submatrix was exactly updated. We can now write: *( ks) ( ks) dpxd * ( ks) ( N dp) xd (9) 8

9 ( k s) Since Q *( k s) and Q * ( k s) are both orthogonal, we can readily see that the submatrix satisfies the equations: * ( k s) * ( k s) ( k s) ( k s) and the error covariance matrix is therefore updated. (30) 9

10 3. MEHOD : ODE UPDAING he data matrix ( p) K at order p can be rewritten as: ( k) y ( k) ( p) ( k ) y ( k ) ( p) K Nx( dpd ) K Nxdp K Nxd ( k N ) y ( k N ) If the model order is updated to p, the data matrix has the form: (3) ( p) ( p) ' K Nx( d ( p) d ) K Nxdp K Nxd K (3) Nxd where ' K are the added d columns Nxd K ' Nxd y ( k ( p )) y ( k ( p ))... y ( k N ( p )) (33) We can then compute the following matrix: ( p) ( p) ( p) ( p) ( p) ( p) ( p) ' ( p) Q K Nx( d ( p) d ) Q K Nxdp Q K Nxd Q K Nxd ( p) 0 where dp d and ( Ndp ) d are extracted from Q ( p ) ' K. (34) We must now triangularize the right term matrix in equation (34). his can be done with a set of Householder transformations or Givens rotations. If we decompose only the small submatrix, it easily yields: Q 0 where dxd is an upper diagonal matrix and Q ( N dp) x( N dp) is the product of the Householder transformations or Givens rotations. (35) Equation (34) then becomes: 0

11 where 0 0 ( p) ( p) I 0 ( p) ( p) dpxdp Q K Nx( d ( p) d ) 0 ( p) 0 Q Q ( p) ( p) Idpxdp 0 ( p) ( p) ' Q K Nx( d ( p) d ) 0 0 Q '' 0 0 ' dxd and are obtained from multiplication '' ( N dp d ) xd ' " Q. (36) (37) It can be seen that the first dxp rows of the right hand side in equation (37) are not affected by the above transformations and the factor matrix ; ( p) ( p ) 0 as was the Q matrix: Q I 0 ( p ) ( p) dpxdp Q 0 Q ; ( p) ( p ) ' ( p) " he covariance matrix of the error was also updated: ˆ ( p) ( p) '' '' Ep ( p ) at order p was thus updated: (38) (39) (40) 4. MEHOD 3: EVESE ODE UPDAING he reverse order updating of the Q factorization is considered from a maximum order p in the same data set. Consider that at sample index t, the data matrix be partitioned to the data matrix term. ( p ) K by removing its last d columns ( p) K of model order p can ' K Nxd of the regressors ( k) y ( k) ( p) ( k ) y ( k ) ( p) ( p) ' K Nx( dpd ) K Nxdp K Nxd K Nxd ( p ) K Nxd K Nxd ( k N ) y ( k N ) K K K (4) ( p) ( p) Nx( dp( p) d ) Nxd ( p) Nxd (4)

12 Since the sample number N is always larger than the data dimension d, the data matrix can have the form: ' '' ' ( p) ( p) ( p) ( p) ( p) ''' '' K Q 0 ( p) Q ( p) ( p) 0 ( p) 0 0 hen we can readily see that: ' ' ' ' ' ' ( p) ( p) '' ( p) '' ( p) ( p) #( p) 0 0 # ( p) ( p) # K Q Q Q Q and through exact Q decomposition: ( p) #( p) ( p) ( p) Q Q (45) ( p) K (43) (44) #( p ) It can be seen that matrix can be found by removing the last d columns from the first ( p) subcolumns of matrix and according to (4), it is not an upper triangular matrix. As a result of this, formulation (44) is therefore not a true Q factorization of the data matrix K ( p ). ( p) ( p ) Fortunately, since the first d( p ) columns of the two matrices K and K are similar, their factors are thus identical in the first d( p ) rows and d( p ) columns. hat means ' ' #( p ) that the submatrices and in matrix are exactly as found in the matrix conduct to the updated model at order ( p ). ( p) ' ' ( p ) ˆ ( ) (46) to he only component that is different between #( p ) and not an upper triangular. Note that the energy of matrix ( p ) K ( p ) lies on the matrix # which is is unchanged, and we can have: ( p) #( p ) ( p) #( p ) ( p ) ( p ) ( p ) ( p ) Q Q Q Q ( p) ( p) ( p) ( p) Since Q Q Q Q I, it can be found that: (47) #( p ) #( p ) ( p ) ( p ) and finally the covariance matrix of error can be exactly updated: ˆ E E (49) '' ( p ) ( p ) ( p ) # # '' ( p) '' '' ˆ( p) dxd ( p) hat means the Q factorization is accurately updated from model order p to model order p. (48)

13 Order updating Order downdating Order updating Compte rendus de la e conférence internationale sur l ingénierie des risques industriels, eims 5. ESULS he three algorithms above can be combined to explore the efficiency of three monitoring routines as schematically shown in Fig.. A sliding window s of 0 samples has been used. P max ime updating t=k t=k+s t=k+s p opt p opt p opt p min : updated model, : solution, p opt : optimal order a) outine, ime updating and order updating ime updating t=k t=k+s t=k+s P max p opt p opt p opt p min : updated model, : solution, p opt : optimal order b) outine, time updating and reverse order updating ime updating t=k t=k+s t=k+s p opt p p opt p opt : updated model, : solution, p opt : optimal order c) outine 3, ime updating and order updating or reverse updating 3

14 Fig. Monitoring routines In the first two routines, an a priori estimation of the model order is required. he solution is updated from p min to p max (Fig. 4-a) or inversely updated from p max to p min (Fig. 4-b) and by time, it is updated only at either p min (Fig. 4-a) or p max (Fig. 4-b). he last case works in two dimensions where the solution can be updated with respect to both time and order from a small intuitive value order. We note that when we combine updating in order and in time, the Q factorization is not the true one but the accurate solution is nevertheless always found. 5. Numerical results Consider a theoretical system of two degrees of freedom ( d.o.f.) as shown in Fig.. Both lumped masses are assumed to suddenly and simultaneously change following a step function as in Fig. 3 and the response data due to an impulsion are plotted in Fig 4 at the sampling frequency of 00 Hz. Modal parameters before and after the change are given in able. Fig. System of degrees of freedom Fig. 3 Mass changing as a step function Fig. 4 ime responses after abrupt change able Modal parameters of d.o.f system Mode Before change After change 4

15 Frequency (Hz) Damping rate (%) Frequency (Hz) Damping rate (%) Figures 5-7 show the result of tracking the model order and the modal parameter identification of the varying system from the three routines when a -60dB NS white noise contaminates the data. Fig. 5 Monitoring by routine Fig. 6 Monitoring by routine 5

16 Fig. 7 Monitoring by routine 3 It is seen that the results are almost identical. Both the natural frequencies and damping rate change can be identified under a transient excitation when no noise is present. Optimal order is found to be constant when the data is stationary and can be adjusted to track changes in the system. In the first two routines, optimal order was abruptly increased around the instant of change even if the window data was progressively transited. his transition was better tracked by the third case with respect to optimal order but not to modal parameters since these later need a transitory time to stabilize the fluctuations in data. 5. Experimental results he third routine was applied to monitor the fundamental frequencies of a real bridge superstructure [7]. he structure was excited by the passing of a heavy truck and the excitation was considered to be random. hree accelerometers were mounted on the middle span. he ambient temporal responses in three transversal, vertical and horizontal directions are plotted in Fig. 8 at sampling frequency of 00 Hz. As can be seen in Figure 9, the optimal model order used for the fitting of data is changing and was monitored at between 4 and 7. he first three modes were clearly identified (Fig 0). It is seen that when the vehicle moves to the middle span, there is a variation of each frequency within the corresponding frequency ranges of 8 to 6 Hz, 0 to 3 Hz and 5 to 8 Hz respectively. he first mode is the fundamental bending mode and its frequency tends to decrease whereas in the two other frequencies, there is an increasing trend. he fourth mode is not clearly indentified. However the variation of damping rates is cumbersome and cannot be identified in an environment that is too noisy (Figure ). 6

17 Fig. 8 Ambient vibration data Fig. 9 Monitoring of the bridge Fig. 0 Monitoring of frequencies Fig. Monitoring of damping 6. CONCLUSIONS A method for monitoring systems variations in the time domain has been presented with the using of the multivariate autoregressive modelling. he solution of the least squares method is updated in both the time and model orders. With the innovative updating of the Q factorization, only the submatrices are updated and the solution is accuratly updated when the order either increases or decreases and can be combined with time updating to provide a very fast and effective procedure for monitoring modal parameters. he results from numerical simulations and experimental real applications on a bridge have shown that the method can be widely applied to monitor natural frequency variations even following an abrupt change. he damping is identified from numerical simulations under a transient excitation when no noise is present, but not from experimental data under a random excitation in a very noisy environment. 7

18 7. EFEENCES. Maia N.M.M and Silva J.M.M, 00. Modal analysis identification techniques. oyal Society. No pp Ewins, D.J., 000. Modal testing: theory, practice, and application. nd ed. Mechanical engineering research studies. Engineering dynamics series 0. Baldock, Hertfordshire, England; Philadelphia, PA: esearch Studies Press. XIII, 56 pages. 3. Wasserman D., Badger D., Doyle. and Margolies L., 974, Industrial Vibration-An Overview, Journal of the American Society of Safety Engineers, 9, Hermans L. and Van der Auweraer H., 999. Modal testing and analysis of structures under operational conditions: Industrial applications. Mechanical Systems and Signal Processing 3(), pp Vu V. H., homas M and Lakis A.A., 006. Operational modal analysis in time domain, Proceedings of the 4 th Seminar on machinery vibration, CMVA, ISBN , Montréal, pp Andersen P., 997, Identification of Civil Engineering Structures using Vector AMA Models, PhD thesis, Aalborg University. 7. Vu V. H., homas M., Lakis A.A. and Marcouiller L., 007. Identification of modal parameters by experimental operational analysis for the assessment of bridge rehabilitation. Proceedings of the nd International Operational Modal Analysis Conference, Copenhaguen, Denmark, Vol, pp Vu V.H, homas M., Lakis A.A. and Marcouiller L Effect of added mass on submerged vibrated plates, Proceedings of the 5 th Seminar on machinery vibration, Canadian Machinery Vibration Association CMVA 07, Saint John, NB, pp homas M., Abassi K., Lakis A. A. and Marcouiller J.L., 005. Operational modal analysis of a structure subjected to a turbulent flow, Proceedings of the 3 rd Seminar on machinery vibration, Canadian Machinery Vibration Association, Edmonton, AB, 0 p. 0. Smail M., homas M. and Lakis A.A., 999. Using AMA methods for crack detection in rotors (in French). Proceedings of the 3 e Industrial Automation Int. conf. (AIAI), Montréal, pp.-.4. Basseville M., 988, Detecting changes in signals and systems - A survey, Automatica, vol.4, no 3, May 988, pp Basseville M., Benveniste A., Gach-Devauchelle B., Goursat M., Bonnecase D., Dorey P., Prevosto M., Olagnon M., 993, Damage monitoring in vibration mechanics: issues in diagnostics and predictive maintenance, Mechanical Systems and Signal Processing, vol.7, no 5, Sept., pp Pandit S. M., 99, Modal and spectrum analysis: data dependent systems in state space. New York, N.Y. : J. Wiley and Sons, 45 p. 4. Vu V.H, homas M., Lakis A.A. and Marcouiller L. A time domain method for modal identification of vibratory signal, Proceedings of the st International Conference on Industrial isk Engineering CII, Montreal, ISBN , 007, pp Ibrahim, S.. and Mikulcik E.C., 977. Method for the direct identification of vibration parameters from free responses. Shock and Vibration Bulletin, (47), pp

19 6. Brown, D.L., Allemang,.J., Zimmerman,.D., Mergeay, M., 979, Parameter Estimation echniques for Modal Analysis, SAE Paper No. 790, SAE ransactions, Vol. 88, pp Vu V.H, homas M., Lakis A.A. and Marcouiller L. October 007, Multi-regressive model for structural output only modal analysis, Proceedings of the 5 th Seminar on machinery vibration, Canadian Machinery Vibration Association CMVA 07, Saint John, NB, pp Hannan E. J., 980. he estimation of the order of an AMA process. he Annals of Statistics, vol. 8 (5), pp: Gang Liang, Wilkes D. M. & Cadzow J. A., 993. AMA Model Order Estimation Based on the Eigenvalues of Covariance Matrix. ransactions on Signal Processing, Vol. 4, No 0, pp: Smail M., homas M. and Lakis A.A., 999. Assessment of optimal AMA model orders for modal analysis, Mechanical systems and Signal Processing journal, 3 (5), pp Smail M, homas M. and Lakis A.A., 999. AMA model for modal analysis, effect of model orders and sampling frequency, Mechanical Systems and Signal Processing, 3 ( 6), pp Kashyap. L., 980. Inconsistency of the AIC ule for estimating the order of autoregressive Models. IEEE ransactions on Automatic Control, AC-5, 980, pp: issanen, J Modeling by shortest data description. Automatica, Vol. 4: A. H. Sayed and. Kailath. A state-space approach to adaptive LS filtering. IEEE Signal Processing Magazine, (3):8--60, July ACKNOWLEDGEMENS he support of NSEC (Natural Sciences and Engineering esearch Council of Canada) through esearch Cooperative grants is gratefully acknowledged. he authors would like to thank Hydro- Quebec esearch Institute for their collaboration. 9. BIOGAPHY V.Hung. Vu is a PhD student at at the Department of Mechanical Engineering, ES in Montreal. His research project involves operational modal analysis of submerged structures in a turbulent flow. M. homas is professor at the Department of Mechanical Engineering, ES in Montreal. He is an expert in structures maintenance, vibration and control. A.A. Lakis is professor at the Department of Mechanical Engineering, Ecole Polytechnique in Montréal. His research interests include random 9

20 vibration and wavelet, plate and shell structures and finite element method, etc. L. Marcouiller is a researcher at Hydro Quebec s esearch Institute. He works on the vibration analysis and simulation of structures and machines. 0