Application of Morlet Wavelet Filter to. Frequency Response Functions Preprocessing

Transcription

1 Applcaon of Morle Wavele Fler o Frequency Response Funcons Preprocessng Ln Yue Lngm Zhang Insue of Vbraon Engneerng, Nanjng Unversy of Aeronauc and Asronaucs Nanjng P.R. Chna ) ABSTRACT Frequency Response Funcons FRFs) play a ey role n he frequency doman modal parameer denfcaon. However, measured npu and oupu daa samples could be very shor and wh heavy nose polluon, for example n flgh fluer es. As a resul, serous varance error wll be generaed n he FRFs esmaon. Ths paper presens a me-frequency de-nosng mehod n FRFs preprocessng based on connuous wavele ransform. Morle wavele s employed o consruc a fler ban, whch s fne mpulse response FIR) lnear phase one mananng phase conssency, o reduce he nose before FRFs esmaon. A modfed Morle base funcon s chosen o mee me frequency resoluon requremen whle applyng ransen excaon. A numercal smulaon s conduced usng GARTEUR arcraf model exced by ransen sgnal wh whe nose added o he smulaed response. The resuls show ha accuracy of he esmaed FRFs s mproved. Idenfcaon error of mode dampng s decreased by 30% comparng wh he resuls obaned from he orgnal sgnal. NOMENCLATURE FRFs f ) frequency response funcons sgnal n me doman W a, b) Connuous wavele ransform coeffcen n me doman L R) ψ a b ) a, b ϕ ) he space of he square negrable complex funcons he wavele base funcon. scale facor me locaon modfed Morle wavele funcon n me doman

2 β Φ ω) ω f ω T W ω, ) FFT F ω) IFFT FIR Φ ω, ) he varable coeffcen of modfed Morle wavele modfed Morle wavele funcon n frequency doman cener frequency me he me resoluon he frequency resoluon he frequency resoluon of he samplng daa. radan frequency oal me of analyss daa connuous wavele ransform coeffcen n Frequency doman Fourer ransform analyss sgnal n frequency doman Inverse Fourer ransform fne mpulse response Fourer ransform of he modfed Wavele GARTEUR an numercal smulaon model maxmal wavele coeffcen W max, ) esmaon + pseudonverse I. INTRODUCTION The dynamc characerscs es of he large complex srucures s very mporan. In mechancal engneerng s he ey for srucure desgn. In aerospace engneerng s an mporan mehod for srucure vbraon conrollng. As well as n cvl engneerng plays he sgnfcan role n he srucure healh dagnoses [1][]. Transen excaon s a wdely useful ool n vbraon dynamc es due o s facly n he feldwor and broad frequency band. By ransen excaon he es frequency band s connuous and broad. Many nrnsc frequences may be esed. Generally, Tes daa are lmed and wh heavy nose polluon, for example flgh fluer es daa [3], he flgh hegh and speed change wh he me. The es sae parameers are no seady so s no mean o remove random error by averagng. The acqured daa should be preprocessed o ge fne frequency response funcons FRFs) by usng advance de-nosng mehod.

3 Comparng wh classcal sgnal preprocessng, he wavele analyss s a more useful ool. Snce wavele s wh fne energy, whch has s energy concenraed n me or frequency o gve a ool for he analyss of ransen, non-saonary or me-varyng phenomenon. Mulresoluon analyss MRA) was orgnally formulaed by Malla [4]. The MRA s wdely used o de-nose by decomposon sgnal n dfference frequency band and hen flerng by hard-hreshold. The fne compacly suppored orhogonal wavele s generally un-symmery excep he Harr wavele or b-orhogonal bases of compacly suppored wavele. The fler phase s nonlnear. The dsoron wll be generaed afer wavele flerng. In addon, he algorhm s complcaed, he flerng effec s no good unless he sgnal band s n low frequency band and he nose s n hgh frequency band. Wh a vew o fner resoluon and lnear phase, non-orhogonal wavele ransforms play an mporan role n sgnal processng by offerng fner resoluon n he choce of waveform, and adjusmen of resoluon n me and scale [5][6]. Morle wavele [7] s expressons s Gaussan funcon n me and frequency doman. I s facly o consruc a FIR fne mpulse response) lnear phase fler as well as a narrowband fler. The advanages of he Gaussan are as follows: 1) The daa characerscs of flgh fluer es are ransen snusodal or pulse, so he selecon of a wavele bass funcon should encompass hs characersc. ) The Guassan s concenraon n me and frequency doman. 3) The smple algorhm n me and frequency doman. So we nvesgae a modfed Morle wavele and s propery and develop a lnear phase fler ban o denose. In Secon II we brefly revew he connuous non-orhogonal wavele ransform. A chosen Modfed Morle wavele are dscussed. Accordng o Hesenberg uncerany prncple we adjus he fner me and frequency resoluon. In secon III we nvesgae he wavele fler ban and s algorhm. In secon IV we conduc numercal smulaon on he wavele flerng mehod. In secon V we presen he de-nose resul based on modfed Morle fler ban. II. OVERVIEW OF NONORTHOGONAL WAVELET ANALYSIS The wavele ransform WT) of sgnal f ) s defned as he nner produc n he Hlber space of f ) L R) norm as follows [8] W a, b) = = a 1/ R f ), ψ a, b ) * b f ) ψ ) d a 1) b L R) s he space of he square negrable complex funcons. where ψ a, b ) =ψ ) s he wavele base a funcon. a s scale facor and b s me locaon, he facor 1/ a s used o ensure energy preservaon. The

4 asers sands for complex conjugae. The wavele analyss nroduced above s called connuous wavele ransform CWT). I s also equvalen o flerng a sgnal hrough a ban of flers ψ ). Snce he flerng s band-pass. Wavele analyses end o concenrae he sgnal energy no relave wavele coeffcens. The general Morle wavele ransform echnque was developed n he 1980s. The general formula s a, b 1 ψ ) = e cos u) ) π In me doman he wavele s cosne funcon modulaed by Gaussan funcon and quc aenuaon. In frequency doman s he ban-pass fler and he Gausson funcon. The properes of me and frequency doman s shown n he Fg. 1 Fg. 1. Real par op), Imagnary par mddle) Fourer ransform of real par boom) of he Morle wavele Enlghened by Gaussan funcons and loong for he less me resoluon o deec he mpulse n me doman we modfed he general Mole wavele o he followng form. β β ϕ ) = e cos ) 3) π When enlargng β, ϕ) s aenuaed o an mpulse sgnal, he me resoluon s fner. When reducng β, ϕ ) s aenuaed o cosne sgnal. The frequency resoluon s fner. A he meanme he dfferen β wh he

5 dfferen wdh of frequency fler. The larger β s, he broader fler wdh wll be. The less β s, he narrower fler bandwdh wll be. For example, Supposng β are 0 and 40, we ge wo Morle wavele funcons, ϕ1 ) = e π cos ) 4) ϕ ) = e cos ) 5) π her correspondng Fourer ransforms, Φ ω) = e 1 ω ) e ω + ) 800 6) Φ ω) = e ω ) e ω + ) 300 7) Fg., Fg. 3 llusraes he dfferen waveform wh dfferen β n me doman and n frequency doman. Fg. Waveform n me doman Fg. 3 waveform n frequency doman Supposng β β ϕ ) = e cos ) = 10 π 5 he number 5 10 a rgh of he equaon expresses ϕ) closes o zero.

6 cos ) 0 Excep o harmonc n horzonal axs s equal o zero.) ln β β ) π = 5 β * ln + 5 π = 8) β The equaon8)s used o selec β accordng o he mpulse duraon. For example, when dong numercal smulaon he mpulse s < s,so β = 0. In addon s he same mporan o deermne he me resoluon and he frequency resoluon ω for he mpulse excaon. The resoluon s lmed by Hesenberg uncerany prncple [9] ω 9) ω and are bandwdh of me wndow and frequency wndow respecvely. For mpulse excaon, daa. So = T π f ω = π s seleced. Hz) s near 1 f = f =. f s he frequency resoluon of he samplng T III. THE WAVELET FILTERING ALGORITHM Based on nvesgang abou modfed Morle wavele, we use a = cons an Morle waveles as a base funcon o consruc fler ban [3]. Accordng o he Parseval equaon,he samplng es daa convolves he correspondng base wavele o produce he wavele coeffcen W ω, ) = FFT < f ) ϕ ) > = F ) Φ ω, ) 10) ω

7 In he frequency doman, he wavele coeffcen s a wavele fler wh cener frequency. Because lnear phase s desred, we only consder he real par of modfed wavele. The wavele coeffcen W ω, ) s real and he reconsrucon s no phase dsoron. The Fourer ransform of he modfed Morle wavele s real par s ω ) β ω+ ) β Φ ω) = e + e 11) u Here he Fourer ransform of he modfed Wavele s Φ ω) ω, ) Φ By he nverse Fourer ransform we produce he me-frequency coeffcens. W, ) = IFFT F ) Φ ω, )) 1) ω F ω) and Φ ω, ) are he Fourer ransform of sgnal and modfed Morle wavele respecvely. When cener frequency s changng n desred analyss bandwdh ω, ) Φ becomes he real Morle fler ban as well as a fne mpulse FIR) lnear phase fler for each. Afer flerng he orgnal sgnal s no phase dsoron. By he equaon 1) he es daa are projeced o a me-frequency analyss plane. In he plane he sgnal energy s concenraed and nose energy s scaered. We can dscard he nose by usng sof-hreshold as follows 1) fndng he maxmal coeffcen W, ). ) he sof-hreshold s seup wh % ~ 3%) W max, ) max 3) he me-frequency wavele coeffcens are se o zero when hey are less han he hreshold or are preserved when hey are grea han he hreshold. Afer sof-hreshold, he me-frequency wavele coeffcen s, W ω ). Thnng abou = 1, L, n, here are n wavele base funcons o be used o buld he FIR fler ban accordng o equaon 11). Supposng a assumpve frequency ω correspondng o a sgnal specrum F ω ), he wavele coeffcens are W ω, 1) Φ 1 ω ) M = F ω ) M 13) W ω, ) Φ ) n n ω

8 The nverse Fourer ransforms of wavele coeffcens n frequency doman are used o ge me- frequency coeffcens elmnaed nose by sof-hreshold mehod. Then he new wavele coeffcens, W ω ) replace he orgnal coeffcens W ω, ) n frequency doman. So a every frequency ω, pseudonverse. F ω ) s solved by he Φ Fˆ ω ) = Φ 1 n + ω ) W ω, 1) M M 14) ω ) W ω, n ) and s nversely ransformed o ge he me doman reconsruced sgnal f ) ). IV. NUMERICAL SIMULATION Wh he bacground of flgh fluer es and consderng he FRFs have he dense mode frequency. In order o valdae he de-nosng mehod, A numercal smulaon wh hree close modes based on GARTEUR arcraf model [10] s conduced whch s exced by pulse sgnal wh whe nose added o he smulaed response. Fg 4 GARTEUR model GARTEUR arcraf model Fg.4 s desgned by he Group for Aeronaucal Research and Technology n Europe GARTEUR) n he medum erm of 90s. FEM consss of 51 beam elemens. Sx dofs for each node. Alogeher here are 68 nodes wh 408 degrees of freedom. GARTEUR arcraf model has 0 modes n 300Hz,Table 1 shows he frs 5 modes.

9 Table1 GARTEUR arcraf modes Mode frequency Mode specfcaon damp %) number Hz) The wng frs bendng Wng an-symmery second bendng and vercal ral frs bendng Wng an-symmery frs orson Wng symmery orson Wng an-symmery second orson Fg. 5 GARTEUR frs 5 modes Fgure 5 shows frs 5 modes sysem FRFs wh hree close modes beween 3Hz o 36Hz wh whch we mae smulaon sysem model. Durng he numercal smulaon 10%,15%,5% whe nose were added respecvely. Mode parameers are denfed usng raonal fracon orhogonal polynomal curve fng mehod [11]. Table GARTEUR hree dense modes sysem parameer denfcaon 10% 15% 5% M Fr. error damp error Fr. error damp error Fr error damp error N Hz) %) %) %) Hz) %) %) %) Hz) %) %) %) B F A F Noae BF s before flerng. AF s afer flerng. MN s mode number. Error s relave error.

10 Table shows ha when he nose level s low 10% whe nose) he accuracy of mode frequency and mode damp denfcaon s almos equal before flerng and afer flerng. Bu when nose s heavy 5% whe nose) he denfcaon accuracy s mproved hghly afer flerng. Especally he damps correspond o he frs and second mode frequency. Relave error of he damp denfcaon s 0% and 44% before de-nosng Afer de-nosng hey are 3.6% and 5.% Fg. 6 shows he smulaon model ha s he heorecal FRFs of hree dense modes sysem. Fg.7s wavele coeffcens flerng compared n he me-frequency plane. I s dsnc ha nose s elmnaed. Fg.8- Fg.9 show he flerng resul of esmaed FRFs n he way of amplude and phase. Fg.6 sysem heorecal FRFs of he smulaon model Fg.7 npu lef),oupu rgh), orgnal op),flered boom) wavele coeffcens Fg. 8 Amplude and phase fgure of BFlef) and AFrgh)

11 Fg. 9 Phase fgure of BFlef) and AFrgh) V CONCLUTION In he processng of lmed and heavy nose pollued dynamcal es sgnal, he me frequency mehod s performed wh non-orhogonal wavele. In he paper modfed Morle wavele s employed o consruc a fler ban, whch s fne mpulse response FIR) lnear phase one mananng phase conssency, o reduce he nose before FRFs esmaon. A modfed Morle base funcon s chosen by adjus he parameer β,, f )o mee me and frequency resoluon requremen whle applyng mpulse excaon. A numercal smulaon s conduced usng GARTEUR arcraf model exced by mpulse sgnal wh whe nose added o he smulaed response. The resuls show ha accuracy of he esmaed FRFs s mproved. Idenfcaon error of mode dampng s decreased comparng wh he resuls obaned from he orgnal sgnal. ACKONWLEGEMENT The auhor of he paper would le o express apprecaon of he suppor from he Naonal Naural Scence Foundaon of Chna NSFC) and Aeronaucal Scence Research Foundaon ASRF) under he research projecs of NSFC # and ASRF #00I5074.Meanme we hans professor M. Ln for provdng he GARTEUR model daa. REFERENCES [1]. Zhang Lngm,Yue ln,xu Qnghua,In-Operaonal Sysem Idenfcaon Theory & Mehods of Large Complex Mechancal Srucures Naonal Naural Scence Foundaon of Chnaconfrmed number )000 []. Zhang Lngm,Xu Qnghua,Yue ln, Idenfcaon Mehod of Flgh Fluer Tes for New Generaon Baleplane, Aeronauc Scence Foundaon of Chna confrmed number:00i5074) 000

12 [3]. Mary Brenner, Wavele Analyses of F/A-18 Aeroelasc and Aeroservoelasc Flgh Tes Daa[C]. 38h AIAA Srucures, Srucural Dynamcs and Maerals Conference. Apr.1997, 696~698 [4]. Malla S. A, Theory for Mulresoluon Sgnal Decomposon: The Wavele Represenaon. IEEE Trans on PAMI 1989,117): 674~693 [5]. Shensa, Mar J., Dscree Inverses for Nonorhogonal Wavele Transforms, IEEE Transacons on Sgnal Processng, vol. 44, no. 4, Apr.1996, pp [6]. Shensa, Mar J., The Dscree Wavele Transform: Weddng he A Trous and Malla Algorhms, IEEE Transacons on Sgnal Processng, vol. 40, no.10, Oc.199,pp [7]. Daubeches, Ingrd, Ten Lecures on Waveles, Socey for Indusral and Appled Mahemacs, Phladelpha, PA, 199. [8]. Charles K Chu, A nroducon o Wavele Xan January, Academc Press 199 pp.3-0 [9]. Leon Cohen, Tme Frequency Analyss: Theory and Applcaons Prence Hall 1995 pp [10]. M.ln, M. Frswell, Worng Group 1: Generaon of Valdaed Srucural Dynamc Models-Resuls of a Benchmar Sudy Ulsng he GARTEUR SM-AG19 Tes-bed, Mechancal Sysem and Sgnal Processng, 171), 003 pp. 9-0 [11]. Fu Zhfang,Vbraon mode analyss and parameers Idenfcaon Peng: Chna Machne Press 1990 pp