Wavelet based recursive identification of modal parameters

Transcription

1 Wavelet based recursve dentfcaton of modal parameters Andrzej Klepka, adeusz Uhl AGH Unversty of Scence and echnology, Department of Mechatroncs and Robotcs, Al. Mckewcza 30, Krakow, Poland tel , fax , emal: {Andrzej.Klepka, Abstract hs paper presents a recursve method of modal parameters dentfcaton based on operatonal measurements, dedcated for non-statonary systems. Decouplng sgnal components procedure allowed to reduce the sgnal model and smplfed the process of modal parameters estmaton. Adaptve method of flterng smplfed the process of wavelet functon selecton. Presented method utlze Contnuous Wavelet ransform (CW) wth Complex Morlet Wavelet functon. hanks to reducton of model order, estmaton of modal parameters can be performed usng relatvely smple mathematcal formula. hs approach sgnfcantly reduces demand for computng power whch have a drect mpact on system costs and modal parameter estmaton tme. he method has been tested on numercal models and appled for real data. Introducton he operatonal modal analyss (OMA) s wdely used n cvl, mechancal and aerospace engneerng communtes and appled to dentfy the modal parameters of such structures as buldngs, towers, brdges, offshore platforms, arplanes, etc. []. However, the classcal OMA s technques have some lmtatons, among whch the followng are the most mportant []: the structure s assumed to be lnear, the structure s tme nvarant, the structure s observable and n the system of nterest dampng s small or proportonal. Due to ths assumptons, results whch can be acheved wth modal technque are an approxmaton of the real structure behavor, but stll, they are good enough to be appled n dagnostcs, montorng, control, etc. In practce, many engneerng structures lke arcrafts, traffc-excted brdges, robots, rotatng machnery workng wth varyng speed, cranes and many others should be treated as non-statonary systems. hs means that at least one of the assumptons s not satsfed n ths case. he classcal technques of OMA do not allow for varaton of the egenvalue matrces [3], [4]. herefore, the technques cannot be drectly appled to dentfcaton of non-statonary systems modal parameters. Addtonally, there are practcal problems assocated wth the mplementaton of the OMA methods ncluded: length of the data requred for analyss, duraton of the estmaton procedure, necessty of an experenced operator s nterventon n order to select the correct results from a set of solutons or nfluence of estmaton procedure sequence n case of data obtaned durng a seres of partal experments (runs or set-ups). he most frequently used soluton for applcaton OMA to dentfcaton of non-statonary system s the quasstatonarty assumpton. It s assumed that the system s statonary wthn gven tme nterval. Unfortunately, ths soluton requres a compromse between the ablty of the algorthm to keep up wth modal parameters changes and the qualty of results (small number of samples). Another approach s the use of recursve methods of dentfcaton. In ths case, the egenvalue matrx can be estmated recursvely for every sample. Despte the above mentoned lmtatons, the OMA technques can be adapted to dentfcaton of nonstatonary systems. Varous varetes of OMA are wdely used n non-statonary condton and can be appled for dentfcaton e.g. systems contanng rotatng parts (such as turbnes, helcopters, engnes, etc.) where the assumpton of broadband of exctaton s not fulflled [5], structures wth varable mass [6] and geometry

2 [7], systems wth closely spaced modes [8], large structures [9] and systems wth varyng boundary condtons [0,]. hs paper focus on applcaton of adaptve approach to wavelet flterng process. he use of wavelet flter allows to: decouplng ndvdual frequency components of the sgnal, reducton of the sgnal model and smplfed the process of the modal parameters estmaton. he adaptve approach smplfed the process of wavelet functon selecton and enables changes/adaptaton of wavelet frequency durng dentfcaton process. he paper s organzed n the followng way. Secton descrbes the method for model parameters estmaton. Secton 3 presents the adaptve wavelet based sgnal fltraton. In Secton 4 the algorthm s presented. he next sectons contans the results of verfcaton of the method on smulated data. Identfed parameters are compared wth the results obtaned by usng a non-adaptve formula of presented algorthm.erreur! Sgnet non défn. Identfcaton method he proposed algorthm conssts of three man parts. In the frst step the sgnal s decomposed by wavelet transform. In second step model parameters are estmated. In thrd step a modal determned. Organzaton of the algorthm shown n Fgure. Fgure : Organzaton of the algorthm. Adaptve wavelet flterng he wavelet analyss s a method of sgnal decomposton. As a result of the wavelet analyss, n contradcton to the Fourer transform, elementary sgnals so called wavelets are obtaned. Wavelet functons are contnuous, oscllated wth varous duraton tmes and spectrums. From the mathematcal pont of vew, a contnuous wavelet transform (CW) of a sgnal x(t) can be defned as xa, b xt * W g g a t b dt a where b s a translaton (dsplacement) representng regon, a s dlataton (expanson) or scale parameter, g(t) s the basc wavelet functon. Wavelet flterng decomposes partcular sgnal frequency components wth resoluton dependng on the wavelet functon parameters []. he selecton of parameters of the wavelet functon requres some compromse between the qualty of fltraton n frequency and tme doman. After decouplng, every component of the sgnal can be analyzed separately. hs sgnfcantly decreases computatonal effort because model order of decoupled component s low. Besdes, trackng all natural frequences and dampng ratos of the system s not always demanded. An example of ths can be flght flutter tests, where very often only a specfed number of frequency and dampng ratos (drectly responsble for the flutter phenomenon) s tracked. It also reduces the demand for computatonal power whch ncreases applcablty of the method and makes the real-tme mplementaton process easer. Schematcally, the process of frequency component decouplng for statonary sgnal s shown n Fgure.

3 Fgure. Wavelet based sgnal flterng. hs property has been repeatedly used to dentfy modal parameters for both statonary and non-statonary systems. he major nconvenence of usng the non-adaptve verson of a wavelet flter for modal parameters dentfcaton process s the constant fltraton bandwdth for assumed wavelet functon. For large changes n system parameters and use of a narrow band flter, decomposton process can be performed for another(next) sgnal component or decomposton can be appled for frequency band where there are no frequency components. hen, the obtaned results can be only a flter response hs problem s presented n Fgure 3a. In turn, applcaton of broadband wavelet flter causes decrease n tme doman resoluton, accordng to Hesenberg relaton. Fgure 3. Comparson of Non-Adaptve (constant scale parameter) (a) and adaptve (varable scale parameter)(b) wavelet flterng concepton. he f and f are tracked frequency, related to scale parameter a. he soluton of the constant flter bandwdth problem can be makng the bandwdth parameter condtonal on the dentfcaton process. hs requres determnaton of the wavelet parameters and dfferent wavelet functons g for dfferent dscrete tme moments : 3

4 W x a, b x t g a Determnaton of the wavelet functons, whch allows fltraton of the gven frequency component, requres defnng the scale parameter assocated wth the frequency by the formula [3]: * t b g dt a f a s where s s the samplng tme and f s the frequency correspondng to scale parameters a. A change of the scale parameter a allows change of the wavelet flter frequency. Schematcally, ths process s shown n Fgure 3b.. Recursve Least Square algorthm Method of estmaton model coeffcents based on RLS algorthm. Schematcally the algorthm s presented n Fgure 4 Fgure 4. Organzaton of Recursve Least Square method. where y () s current system response sgnal, ˆ ( ) s estmated a pror predcton error, based on prevous teraton ˆ ( ) y( ) ( ) ˆ( ) () s regressor vector, () matrx gven as s vector of model parameters, L s gan vector and P( ) ( ) L( ) ( ) P( ) ( ) P s covarance P( ) ( ) ( ) P( ) P( ) P( ) ( ) P( ) ( ) where - forgettng factor. For, = 0, I - unt matrx. P( 0) I,where - s a large natural number, for example 0 6, 4

5 As a result of RLS algorthm the vector of model parameters s obtaned ( k) [ a,..., a,, c,..., c n A nc ] where n A and n C are model order, a,..., a, c,..., c n n are model coeffcents. A C 3 he algorthm he adaptve dentfcaton algorthm conssts of two man parts. he core of the algorthm s RLS algorthm responsble for model parameters estmaton. he nput data are fltered usng wavelet transform, whch allows both reduce the model order and estmate the modal parameters based on analytcal formulas. An ntegral part of the dentfcaton process s an adaptaton of wavelet flter that allows to tune the flter characterstcs to the current value of the frequency. 3. he adaptaton process he adaptaton process s performed by comparng the current frequency of wavelet functon (a parameter) and frequency estmated by the RLS algorthm. If the absolute value of the dfference of the two frequences s contaned wthn the assumed range (δ), the dentfcaton process s contnued wthout changes. If the dfference s greater than assumed, the frequency of wavelet functon (scale parameter) s changed to a value correspondng to the frequency estmated from the RLS algorthm. Schematcally, the process of adaptaton and the dagram of the method wth adaptve wavelet flterng s presented n Fgure 5, where f e, f w and δ are respectvely: current estmated frequency, frequency correspondng to scale parameter (wavelet frequency) and adaptaton step. he adaptaton step allows to reduce the nstantaneous growth of covarance matrx values at the moment of change of the wavelet flter parameters [4,5]. he range of ths parameter s determned as much smaller than the flter bandwdth (-5% of bandwdth) to guarantee the correct sgnal components decouplng. Fgure 5. Organzaton of proposed adaptve wavelet flterng procedure. 3. Modal parameters estmaton Wavelet decouplng allows to separaton partcular sgnal components. he result s that model order of the sgnal s known and equal two. For the second order sgnal model t s possble to assgn analytcal formulas that descrbe dependences between model and modal parameters [5] ln a a 4a arctan a a 4a s s 5

6 s ln s ln a a 4a a a 4a arctan a 4a s, a where: - natural frequency, - dampng rato. When the analytcal formula for modal parameters calculaton s assgned, there s no requred to fnd roots of the characterstc polynomal estmated from the RLS algorthm. 4 Verfcaton of the algorthm numercal non-statonary model he two degree of freedom non-statonary system was modelled. he stffness coeffcent for mode was changed durng smulaton accordng to equaton An example of tme hstory of the system response and the stffness parameters are presented n Fgure 6. Fgure 6: a)system response for whte nose exctaton, b) stffness changes for mode Usng procedures descrbed n the secton 3, the dentfcaton process was performed. Both adaptve and non-adaptve algorthms had the same ntal parameters appled (wavelet functon and forgettng factor). Wavelet functon parameters were selected randomly. Comparsons of the dentfcaton results usng nonadaptve and adaptve flterng are presented n Fgure 7 where dentfed values of dampng rato and natural frequences of the system are presented. 6

7 Fgure 7. Comparson of dentfcaton results for mode : a) natural frequency, b) dampng rato As can be notced, the method wth adaptve wavelet flter gves better results for both natural frequency and dampng rato. here are two conclusons arsng from the performed test: adaptve wavelet fltraton enables trackng of modal parameters and the process of ntal wavelet selecton s not a crtcal part of the algorthm, unlke n the non-adaptve method (as descrbed n prevous work of the authors). 5 Expermental verfcaton of formulated procedures wo experments on real objects were performed. Frst, the real tme dentfcaton of the modal parameters of a system wth varable stffness has been conducted. Next, the algorthm has been used for dentfcaton of the modal parameters of ISKRA ar jet durng a flght. 5. Identfcaton of modal parameters of system wth varable stffness. he test bed was buld out of the three man parts: frame, cart and two metal bellows. Between the cart and the frame there are two metal bellows mounted. he frcton between the cart and the frame has been elmnated thanks to ar bearngs. Expermental setup of experment s presents n Fgure 8. Fgure 8 : Experment arrangement: a) scheme of the laboratory stand, b) equpment for experment prepared to free vbraton run test. frame,. cart on the ar bearngs, 3. ar bearng hoses, 4. metal bellows bracket mounted to the frame, 5. ar bearngs, 6. metal bellows bracket mounted to the sldng cart, 7. two metal bellows. 7

8 An electromagnetc shaker and a sgnal generator were used to excte the structure. he whte nose sgnal was used as an exctaton sgnal. Durng the experment, the pressure n metal bellows was changed. As a result of system stffness changes, the natural frequency of the system was shfted. he tme hstory of system response, the result of the natural frequency dentfcaton and the adaptaton process of the wavelet functon are presented n Fgure 9. Addtonally, wavelet adaptaton process was presented n Fgure 9b (dashed lne). Fgure 9. System response (a), Comparson of dentfed natural frequency of the system (b). Also n ths case, the results obtaned wth the use of adaptve methods are consderably better than the ones obtaned by the non-adaptve wavelet flterng. he algorthm reacts much faster to the changes of the natural frequency of the system. Both algorthms were run wth the same ntal parameters. 5. Modal parameters dentfcaton of Iskra ar jet durng a flght. S- Iskra s a two-seater, md-wng monoplane jet. he study of the n-flght modal characterstcs of the arcraft was based on ten accelerometers, arranged as shown n Fgure 0a. Durng the flght test the arcraft was accelerated to a speed exceedng the maxmum speed (durng the dve). An example of a system response acqured by sensor 7 s presented n Fgure 0b. Fgure 0. In-flght test: a) Measurement ponts for n-flght test, b) Example of response sgnal for sensor 7. 8

9 Analyss was perform for mode 7Hz whch has the bggest nfluence on the generaton of flutter phenomenon. he comparson of the results estmated by real-tme adaptve algorthm wth the results estmated by off-lne classcal non-adaptve RLS algorthm wth band-pass flterng s presented n Fgure. Fgure. Comparson of result: a) dampng rato, b) frequency he comparson of the results shows that both algorthms gve smlar results wth excepton of the ntal phase of dentfcaton process (Fgure b) where dampng rato value s relatvely large. It s worth mentonng that the results of the classcal RLS algorthm were obtaned n off-lne mode, wth band-pass fltraton. 6 Conclusons Applcaton of the adaptve wavelet fltraton to recursve dentfcaton of modal parameters have been nvestgated. he performed test confrmed that the wavelet transform s a useful tool to support the dentfcaton process of a non-statonary systems. he adaptve wavelet fltraton allows to separate the sgnal frequency components and enables reducton of model order of analyzed sgnal. hs approach sgnfcantly reduce the computaton tme of modal parameters thanks to the use of analytcal formulas for dampng rato and natural frequences and facltate the hardware mplementaton of the algorthm. he algorthm also allows to determne the confdence ntervals of modal parameters, whch gves the possblty to assess the qualty of results. All performed tests showed that the adaptve wavelet fltraton method combned wth the RLS algorthm gves satsfactory results and works much better than non-adaptve verson of the algorthm. hs does not dsqualfy the non-adaptve formula of the algorthm. In the authors prevous works ths non-adaptve approach and ts hardware mplementaton was successfully used for real-tme dentfcaton of modal parameters of non-statonary systems [4; 5] and enable to track tens of natural frequency n real-tme (assumng maxmum samplng frequency of the sgnal equals 00Hz ). In ths case the process of selectng the ntal wavelet functon and the forgettng factor had to be performed very precsely. Acknowledgments addtonal nformaton References [] Uhl., Lsowsk W.,Kurowsk P. In-operaton modal analyss and ts applcaton. Krakow : AGH Unversty of Scence and echnology, 00. [] Persol Allan G Paez homas L. Harrs' shock and vbraton handbook. New York : McGraw- Hll, 00. 9

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