Automated Frequency Domain Decomposition for Operational Modal Analysis

Transcription

1 Autoated Frequency Doain Decoposition for Operational Modal Analysis Rune Brincker Departent of Civil Engineering, University of Aalborg, Sohngaardsholsvej 57, DK-9000 Aalborg, Denark Palle Andersen Structural Vibration Solutions A/S Niels Jernes Vej 10, DK-9220 Aalborg East, Denark Niels-Jørgen Jacobsen Brüel & Kjær Sound & Vibration Measureent A/S Skodsborgvej 307, DK-2850 Næru, Denark Noenclature t,τ ie f Frequency y (t) Syste response N Nuber of easureent channels φ, Φ Mode shape, ode shape atrix C G U Covariance atrix Spectral density atrix u, Singular vector, Matrix of singular vectors [ s i ] Diagonal atrix of singular values d 1, d 2 Discriinator functions Ω 1,Ω 2 hreshold levels µ, σ, γ Mean, standard deviation, kurtosis Abstract he Frequency Doain Decoposition (FDD) technique is known as one of the ost user friendly and powerful techniques for operational odal analysis of structures. However, the classical ipleentation of the technique requires soe user interaction. he present paper describes an algorith for autoated FDD, thus a version of FDD where no user interaction is required. Such algorith can be used for obtaining a default estiate of odal paraeters in coercial software for operational odal analysis - or even ore iportant it can be used as the odal inforation engine in a syste for structural health onitoring. Introduction Frequency doain techniques have always been popular. Even aong people who will state that they only use tie doain techniques for odal identification, as soon as they get new data in their hands, the first thing they will norally do is to take a look at soe frequency doain functions. For Operational Modal Analysis frequency doain techniques are based on spectral density functions, Bendat and Piersol [1].

2 Working directly with spectral density function has been popular and is still used a lot; see for instance Felber [2]. One of the probles working directly with the spectral density functions is the aount of data the user has to work with siultaneously. For instance in a case with N = 8 channels of easureents, the user has to deal with 2 (8 + 8 ) / 2 = 36 different spectral density functions. Further, even though the spectral densities by directly depicting the odal peaks and thus gives a direct indication of the presence of odes, the spectral density function in itself does not provide the user with odal inforation since the spectral density function is linear cobination of the odal responses. herefore working directly with spectral density functions will liit odal identification to cases with well separated odes. he Frequency Doain Decoposition technique is a way to solve these two probles, Brincker et al [3], [4]. he technique siplifies the user interaction because the user has only to consider one frequency doain function - the singular value plot of the spectral density atrix. his plot concentrates inforation fro all spectral density functions. Further, if soe siple assuptions are fulfilled, the technique directly provides a odal decoposition of the vibration inforation, and the odal inforation for each ode even in the case of closely spaced odes and noise can be extracted easily and accurately. he principle in the Frequency doain Decoposition (FDD) techniques is easiest illustrated by realizing that any response can by written in odal co-ordinates Now obtaining the covariance atrix of the responses y t) = φ q ( t) + φ q ( t) +... = Φq( ) (1) C ( t { y( t τ ) ( t } ( τ ) = E + y ) (2) and using equation (1) leads to hen by taking the Fourier transfor C ( τ ) = E = ΦC { Φq( t + τ ) q( t) Φ } qq ( τ ) Φ (3) qq G ( f ) = ΦG ( f ) Φ (4) hus if the odal co-ordinates are un-correlated, the power spectral density atrix of the odal coordinates is diagonal, and thus, if the ode shapes are orthogonal, then Eq. (4) is a singular value decoposition (SVD) of the response spectral atrix. herefore, FDD is based on taking the SVD of the spectral density atrix [ L [ s ] ( f i S qq ( f ) G ( f ) = U( f ) U ) (5) he atrix U = u 1, u 2, ] is a atrix of singular vectors and the atrix [ s ] i is a diagonal atrix of singular values. As it appears fro this explanation, plotting the singular values of the spectral density atrix will provide an overlaid plot of the auto spectral densities of the odal coordinates. Note here that the singular atrix U = [ u 1, u 2, L ] is a function of frequency because of the sorting process that is taking place as a part of the SVD algorith. A ode is identified by looking at where the first singular value has a peak, let us say at the

3 frequency. his defines in the siplest for of the FDD technique - the peak picking version of FDD - the odal frequency. he corresponding ode shape is obtained as the corresponding first singular vector 1 in. φ = u 1 ( ) (6) u U Introducing odal discriination he process of findings peaks on a function is actually easy to autoate. However, we need to define indicators that can help us distinguishing between different odes and between odes and noise. Let us say that we have identified a peak in the first singular value. he question is now if this is a liable odal peak or is if it just a noise peak. Calculating the correlation between the first singular vector at the peak the ode shape vector at that point - and the first singular vector at neighboring points defines the discriinator function called the odal coherence d 1( f 0 1 u1 ) = u ( f ) ( ) (7) If the odal coherence is close to unity, then the first singular value at the neighboring point correspond to the sae odal coordinate, and therefore, the sae ode is doinating. his function is helpful in discriinating between points doinated by odal inforation and points doinated by noise. If the coponents of each of the vectors in Eq. (6) are rando, then and since the length is unity E { f ) u ( f )} 0 u 1 ( 0 1 = (8) { 1 ( ) u1( f )} = 1 N Var / u (9) hus the ore easureent channels we have the closer two points with rando (non-physical) inforation will get to zero. A reasonable criterion for accepting the neighboring point as a point with siilar physical inforation, and thus accepting the presence of physical inforation at that frequency, could be by introducing a threshold level Ω 1 and the requireent Ω n d 1 Ω 1 (10) setting the liit 1 equal to a nuber ties the standard deviation of the correlation for rando vectors as given by Eq. (9) Ω 1 = n / N (11) where n could be chosen in the region 3-5. his criterion is strongly dependent upon the nubers of easured channels N, if we choose n = 3 and N = 16, then Ω 75, if N = 10, then Ω 1 = hus for 1 = 0. channel counts lower than say 16, the criterion becoes of less value using only correlation between two points. In this case several points on each side of the peak can be cobined to calculate the correlation between the considered peak point and a set of neighboring points increasing the effective value of correspondingly. Once a peak has been accepted as representing odal inforation, another discriinator function can be helpful in discriinating between different odes. In this case the discriinator function is defined as N

4 [ ] s i f 1 d f ν Ω f 1 f2 f Figure 1. Illustration of the definition of the odal doain for of a considered ode. he top picture shows the odal decoposition using the SVD of the spectral density atrix. Botto picture shows how the odal doain is defined by the part of the discriinator function d 2 Ω2. f ν f Figure 2. Exaple of odal discriination. op: SVD of two closely spaced odes easured in two channels. Middle: odal coherence function d 1, botto: odal doain function d 2

5 d ( 2 1 u1 f ) = u ( f ) ( ) (12) hus this discriinator function is not a function of the initial point given by the frequency f0, but is a function of the frequency f of the considered neighboring point. If a high correlation is present over a certain frequency range around the considered peak it eans that over that frequency range only that ode is doinating and introducing a siilar criterion [ ] d 2 Ω 2 (13) defines a frequency range f0 f1; f0 f2 around each peak of odal doinance called the odal doain, see Figure 1. he lower the value Ω 2, the larger the size f = f1 + f2 of the corresponding odal doain. An exaple of discriinator functions are shown in Figure 2. Introducing haronic discriination One iportant proble often arising in practice is when haronics are present in the signal. A haronic is easily confused with a odal peak if not special easures are taken to avoid istakes. he reason is that a haronic appear as a narrow peak in the spectral density functions, thus the peak will also be present in the singular values. he best way to discriinate haronics is by the statistical characteristics of the response in a narrow frequency band around a haronic peak. It is well known that the statistical properties of a haronic are very different fro the properties of a stochastic response. Due to the central liit theore, and the fact that in practice a structure is loaded by any stochastically independent forces, the stochastic distribution of a odal response will be close to Gaussian. Further, the distribution of a haronic is very different fro Gaussian since it has two distinctive peaks where the distribution goes to infinity, see Bendat and Piersol [1], see Figure 3. his difference between stochastic and haronic response was proposed as a basis for haronic discriination in Brincker et al [5]. Figure 3. Noralized PDF of the response of a pure structural ode (left) and pure haronic coponent (right) In Jacobsen et al [6] it is shown how to use the kurtosis to discriinate between odal peaks and haronic peaks. he kurtosis γ of a stochastic variable x provides a way of expressing how peaked or how flat the probability density function of x is. he kurtosis is defined as the fourth central oent of the stochastic variable x noralized with respect to the standard deviation σ as follows

6 γ ( µ, σ ) E 4 {( x µ ) } x = (14) 4 σ Often the nuber 3 is subtracted fro equation (1) as this gives a kurtosis of zero, when x follows as noral distribution γ * ( µ, σ ) = γ ( x µ, σ ) 3 x (15) Using Eq. (15), a PDF with a positive kurtosis is said to be leptokurtic. If its kurtosis is negative, it is said to be platykurtic. A PDF with kurtosis equal to zero is called esokurtic. Leptokurtosis is associated with PDFs that are siultaneously peaked and have fat tails. Platykurtosis is associated with PDFs that are siultaneously less peaked and have thinner tails. For the response of a pure structural ode, the PDF will be norally distributed, and hence the kurtosis γ * = 0 (esokurtic). For a sinusoidal coponent γ * = -1½. his fact is used in the haronic detection technique described further in Jacobsen et al [6]. FDD autoated he search set includes all points on the first singular value plot that is within a predefined frequency band (as a special case the total frequency band of the vibration data) and is above a predefined excitation level. A procedure can be the following 1. Identify a peak on the first SVD representing a axiu 2. check if the peak is likely to be physical 3. If so, establish the odal doain 4. If not define a noise doain around the peak 5. Exclude the odal doain or noise doain fro the search set 6. Continue until the search set is epty, the peak is below the predefined excitation level, or a specified nuber of odes has been estiated he key point of the algorith is point 2). As described earlier, it is essential at this point to include a criterion concerning the correlation between neighboring points as described by the odal coherence function d 1. Also it is essential to be able to distinguish between a haronic peak and a odal peak. Additional criteria can be based on for instance the size of the odal doain being larger than a certain value or the daping estiate being below a certain value. Calculating the daping it ight be useful to isolate the odal coordinate by using the a odal filter as proposed by Zhang [7], thus the auto spectral density for the odal coordinate is calculated by { φ G ( ) φ} Gqq ( f ) = Re f (16) And then the daping is extracted fro the decay of the corresponding auto correlation function. When doing autoated identification quite often we have a priori inforation about what ode we are looking for. In this case we siply use the a priori ode shape φ in stead of u ) when calculating the odal coherence function d 1. If we are looking for a certain nuber of odes, we can take the odes that have the largest odal doain, or could take the odes that have the largest excitation level. For peak picking FDD we are satisfied when the peak is identified as a odal peak, and the corresponding ode shape vector is estiated. For enhanced FDD the following step is to estiate the auto correlation function for the odal coordinate as described above, see ore inforation in Brincker et al [3],[4]. he correlation level Ω used for estiation of the 1 (

7 part of single degree of freedo bell function that is going to be used in the inverse Fourier transfor can be equal to Ω 2, or a different value for Ω can be used. Identifying the daping and natural frequency fro the corresponding correlation function is easily autoated, since a robust algorith is based on excluding soe points in the beginning of the correlat ion function and using only the function down to a certain decay level, for instance using fro 0.95 down to 30 % of the envelope. Conclusions he proposed algoriths have been tested on different data. he conclusion is that the proposed technique is useful and robust and in any cases provides inforation siilar to what can be achieved by anual interaction. References [1] Julius Bendat and Allan G. Piersol: Rando Data, Analysis and Measureent Procedures. John Wiley & Sons, New York, [2] Felber, A.J. : Developent of a Hybrid Bridge Evaluation Syste. PhD. thesis, Departent of Civil Engineering, University of British Colobia, Vancouver, Canada, 1993 [3] R. Brincker, P. Andersen, L. Zhang: Modal Identification Fro Abient Responses Using Frequency Doain Decoposition. In Proceedings of he 18th International Modal Analysis Conference (IMAC), San Antonio,exas, pp , [4] Rune Brincker, Lingi Zhang and Palle Andersen: Modal identification of output-only systes using frequency doain decoposition, 2001 Sart Mater. Struct. 10, [5] R. Brincker, P. Andersen, N. Møller: An Indicator For Separation Of Structural and Haronic Modes In Output-Only Modal esting. Proceedings of he 18th International Modal Analysis Conference (IMAC), San Antonio, exas, pp , [6] N-J. Jacobsen, P. Andersen, R. Brincker: Eliinating the Influence of Haronic Coponents in Operational Modal Analysis. Proceedings of he 25th International Modal Analysis Conference (IMAC), Orlando, Florida, [7] Lingi Zhang, ong Wang and Yukio aura: A Frequency-Spatial Decoposition (FSDD) echnique for Operational Modal analysis. In Proceedings of the 1 st International Operational Modal Analysis Conference (IOMAC), April 26-27, 2005, Copenhagen, Denark.