Vibration Input Identification using Dynamic Strain Measurement

Transcription

1 Vbaton Input Identfcaton usng Dynamc Stan Measuement Takum ITOFUJI 1 ;TakuyaYOSHIMURA ; 1, Tokyo Metopoltan Unvesty, Japan ABSTRACT Tansfe Path Analyss (TPA) has been conducted n ode to mpove the nose and vbaton qualty of mechancal stuctues. Howeve, the foce dentfcaton n the TPA s stll challengng poblem and ts accuacy has to be mpoved. The Matx Inveson Method and the Appaent-Mass Matx Method ae appoaches fo foce dentfcaton. The Matx Inveson Method estmates the exctaton foce by the poduct of an nvese matx of acceleance and a vecto of actual opeatonal acceleaton. It s known that the Matx Inveson Method s vey senstve to measuement nose especally at the esonance. Appaent-Mass Matx Method has been ecently poposed, whch povdes moe accuate esults than the Matx Inveson Method. Howeve, these methods ae stll nsuffcent n accuacy, and mpovement of dentfcaton accuacy s equed. Ths pape poposes a new foce dentfcaton method by usng stan measuement. The method estmates the foce usng the Stan Fequency Response Functon (SFRF) nstead of acceleaton o Appaent-Mass Matx. It s shown that SFRFs ae moe stongly affected by hghe-ode modes than acceleance FRF. Theefoe, we consdeed that moe accuate foce dentfcaton can be obtaned by usng SFRF than the conventonal method. Keywods:Tansfe Path Analyss, Foce Identfcaton, Matx Inveson Method, Appaent Mass Method, Stan, I-ICE Classfcaton of Subjects umbe(s): ITRODUCTIO In ecent yeas, the technques fo educng the nose and vbaton have become mpotant fo mpovement of the poduct s pefomance. In ode to fnd effectve countemeasues, the contbutons of the dentfed souces ae evaluated. Especally, the tansfe path analyss (TPA) has been establshed fo vehcle nose and vbaton poblems. The TPA s a technque fo calculatng contbutons to the evaluaton esponse. In the TPA analyss, the taget system s sepaated nto actve system and passve system. The actve system geneates an nteface load, and the passve system eacts to the nteface load. And each contbuton s calculated as a poduct of exctaton foce specta and fequency esponse functons. Hence, the accuacy of the nput foce dentfcaton nfluences TPA pecson. The exctaton foce needs to be measued by a load cell, but n many cases t s not possble to place a load cell n a pope manne due to space lmtaton o dffculty of keepng bounday condton. Theefoe, It s necessay to conduct nput dentfcaton usng ndect measuements such as a matx nveson method to obtan nput. The matx nveson method estmates the exctaton foce by the poduct of an nvese matx of acceleance and a vecto of actual opeatonal acceleaton. Howeve, measuement eos n the acceleance matx and acceleaton vecto ae popagated to the dentfed foce specta though the nveson pocedue 1. In pevous studes, an appaent-mass matx method has been poposed. In the appaent-mass matx method an appaent mass matx coespondng to an nvese matx of the acceleance s dectly estmated. Hence, nveson of the matx s not necessay n estmaton pocess. Consequently, the appaent-mass matx method povdes moe accuate esults than the matx nveson method. Although the esult mpoved, mpovement of dentfcaton accuacy s stll equed. Ths pape nvestgates a new foce dentfcaton method by usng stan measuement. The method estmates the exctaton foce usng stan fequency esponse functons nstead of the acceleance matx o the appaent mass matx. It s undestood that the ll-condton of the fequency esponse 1 tofuj-takum@ed.tmu.ac.jp yoshmu@tmu.ac.jp Inte-nose 14 Page 1of 8

2 Page of 8 Inte-nose 14 functon matx deteoates the accuacy of the nput dentfcaton. It s expected that f at a taget fequency the fequency esponse functon eceves an nfluence of hgh ode modes, ll-condton of the fequency esponse functon matx s allevated. In pevous studes, t s suggested that stan fequency esponse functons ae moe stongly affected by hghe-ode modes than the acceleance 3,4. Howeve, a clea eason fo t has not been poven n ths study. In ths pape, numecal smulatons ae conducted to compae the esults by the poposed method wth the conventonal method.. Theoy.1 Matx Inveson Method In the matx nveson method, fequency esponse functons H, esponse dung the opeatng condton y and exctaton foce specta f ae elated by the followng expesson. y1 H1,1 H1, m f1 (1) yn H n,1 H n, m f m Whee f j s the foue spectum of the exctaton foce at pont j, y s the foue spectum of the esponse at pont, H, j s the fequency esponse functons wth nput pont j and esponse pont. Rewtng Eq.(1) gves y H f () If a numbe of exctaton ponts m and a numbe of esponse ponts n ae the same (m = n), foce specta ae dentfed by pemultplyng the nvese of the fequency esponse functon matx H to the acceleaton vecto as follows. 1 f H y (3) On the othe hand, n ode to mpove dentfcaton accuacy of foce specta, t s common that the numbe of esponse ponts takes moe than the numbe of exctaton ponts(m < n). In ths case, the exctaton foce s dentfed by the least squaes method. And ths condton s desable to ncease the dentfcaton accuacy. Then the exctaton foce s dentfed as follows. f H y (4) H 1 H whee H s pseudo-nvese matx H H H Η.The supescpt H ndcates the conjugate tanspose. The exctaton foce s estmated by usng the Eq.(3) o Eq.(4). Appaent-Mass Matx Method In the appaent-mass matx method, the exctaton foce s dentfed as follows f G y (5) whee G s the appaent-mass matx. The appaent-mass matx estmaton s the same as H estmaton of FRF when the numbe of esponses ae equal to the numbe of nput foces 5. The dentfcaton accuacy s mpoved by avodng calculaton of the nvese matx. And the appaent-mass matx method estmates the nput foce so that the eos of the nput s mnmzed. Theefoe, the appaent-mass matx method s moe advantageous than the matx nveson method to nput dentfcaton..3 Fequency Response Functons of the Beam In ths pape, the numecal smulaton s conducted to compae the esults by the poposed method wth the conventonal method. The taget stuctue s a beam suppoted at both ends as shown n Fg.1. When consdeng the bendng vbaton of the both-ends suppoted beam, the acceleance (FRF) and stan fequency esponse functons (SFRF) ae expessed as follows. n W xw x H (6) j Al W x 1 x sn (7) l Page of 8 Inte-nose 14

3 Gan[(m/s )/] Gan[[-]/] Phase[deg] Phase[deg] Gan[(m/s )/] Gan[[-]/] Phase[deg] Phase[deg] Inte-nose 14 Page 3 of 8 H z n 1 l W x W x j Al x W x sn (9) l Eq.(6) s the FRF, and the Eq.(8) s the SFRF, whee W and W ae a dsplacement mode functon, and a stan mode functon, espectvely. The stan mode functon s obtaned by dffeentatng the dsplacement mode functon twce wth espect to x., z and n ae a mode ode, a dstance fom the neutal axs to the suface n the bendng of the beam, and numbe of adopted modes, espectvely. A numecal smulaton s conducted to compae FRF wth SFRF. The esponses ae calculated by changng the numbe of adopted modes n(n=5,1,5). The fequency ange s fom [Hz] to 8 [Hz].The bode dagams of the FRF and SFRF (Dvng Pont Responses) ae shown n Fgs. and 3, espectvely. The dagams of Coss Responses ae also shown n Fgs.4 and 5. These esults ndcate that fequency esponse functons ae affected by hghe-ode modes n the case of dvng pont FRF. Especally, the SFRF ae moe stongly affected by hghe-ode modes than the FRF. (8) Fg. 1 Both ends Suppoted Beam n=5 n=1 n=5 5dofs 1dofs 5dofs 1-8 n=5 n=1 n=5 Fg. FRF (Dvng Pont Resp.) Fg.3 SFRF (Dvng Pont Resp.) -18 5dofs dofs 5dofs n=5 n=1 n=5 1-5 n=5 n=1 n=5 Fg.4 FRF (Coss Resp.) 1-8 Fg.5 SFRF(Coss Resp.) 3. umecal Smulaton The numecal smulaton s conducted to compae the nput dentfcaton esults by the poposed method wth the conventonal method. 3.1 Pocedue of umecal Smulaton The numecal smulaton was pefomed as the followng pocedue. Inte-nose 14 Page 3 of 8

4 Page 4 of 8 Inte-nose 14 (a)pepae the exctaton foce, whch s a andom sgnal that follows the nomal dstbuton wth mean and standad devaton 1. (b)obtan the esponse data (acceleaton o stan) by multplyng fequency esponse functons (FRF o SFRF) to the exctaton foce geneated n (a) (c)put eos to the exctaton foce and esponse data(acceleaton o stan)obtaned n (a) and (b),and estmate fequency esponse functons (FRF o SFRF) by egadng them as measued data. The eo data s pepaed as follows: whee y eo, E a eo y k e k / k k / 1 y k y y y s a tue esponse vecto; y s a esponse obtaned n 1 E E e e s an eo vecto; k e k k k e j e s a andom complex numbe wth mean and standad devaton 1 n the k-th measuement at the pont ; e e, m, the eal and magnay pats, espectvely; a s an eo adjustment facto; and s an aveage eo numbe fo nput dentfcaton. (d) Calculate the opeatonal data wthout nose: the exctaton foce and esponse data (acceleaton o stan). Put the eo to the esponse data (acceleaton o stan). (e) Identfy the exctaton foce usng fequency esponse functons (FRF o SFRF) obtaned n (c) and the esponse data (acceleaton o stan) n (d). (f) Evaluate the nput dentfcaton accuacy. The fequency aveaged eo s an eo ndex value by aveagng the dentfcaton eo at all fequences. The fequency aveaged eo s calculated as follows. k k fˆ f M 1 k 1 ave. (11) M u 1 k f k 1 whee k fˆ s the foue spectum of the exctaton foce that have been dentfed n the k-th fequency; k f s the foue spectum of the tue value of the exctaton foce; and M s a numbe of fequency lnes. (1) 3. Condtons of umecal Smulaton Locaton of the esponse ponts and exctaton ponts ae summazed n Table 1. Fo Case A, the esponse data ncludes the dvng pont esponses, wheeas fo Case B, they ae not ncluded. Othe condtons ae as follows: the fequency ange s fom [Hz] to 8 [Hz]; a fequency esoluton s.5 [Hz]; all modal dampng atos ae equal to 1%; and the numbe of adopted modes s 5. Two uncoelated foces ae appled to the system smultaneously. Fequency esponse functons ae estmated wth 5 aveages. The nput dentfcaton s teated 5 tmes. The eo adjustment facto s.1. Table1 Response and Exctaton Condtons Case Response Pont[m] A B Acceleaton Stan Acceleaton Stan Exctaton Pont[m] f1 f Results The dentfed and tue foce powe specta by matx nveson method (M.I.) ae shown n Fg.6. Page 4 of 8 Inte-nose 14

5 Fequency Aveaged Eo[ ] Fequency Aveaged Eo[ ] Powe Spectum[ ] Powe Spectum[ ] Powe Spectum[ ] Powe Spectum[ ] Inte-nose 14 Page 5 of 8 Those by the appaent-mass matx method (A.M.) ae shown n Fg.7. The fequency aveaged eo s shown n Fgs.8 and 9. Fo M.I. case, Case A gves bette dentfcaton esults athe than Case B; and esults of SFRF gves bette esults n Case A but vce vesa n Case B. Fo A.M. Case A gves bette esults than Case B although the dffeence s small; and SFRF gves bette esults n Case A but not vald n Case B. Futhemoe, the appaent-mass matx method (A.M.) gves much bette esults than the matx nveson method (M.I.). 1 - M.I. FRF M.I. FRF M.I. SFRF M.I. SFRF Tue 1 - Tue (a) Case A (b) Case B Fg.6 Foce Identfcaton(M.I.) A.M. FRF A.M. SFRF Tue A.M. FRF A.M. SFRF Tue 1 - (a) Case A Fg.7 Foce Identfcaton(A.M.) 1 - (b) Case B 4 3 M.I. FRF M.I. SFRF A.M.FRF A.M.SFRF Case A. Case B Fg.8 Fequency Aveaged Eo(M.I.) Case A. Case B Fg.9 Fequency Aveaged Eo(A.M.) Inte-nose 14 Page 5 of 8

6 Condton umbe[-] Condton umbe[-] Condton umbe[-] Page 6 of 8 Inte-nose Dscusson 4.1 Condton umbe We compae the condton numbes of the SFRF and FRF matces. The condton numbe s the facto that evaluates the eo popagaton n the exctaton foce that have been dentfed. The condton numbeh s calculated as follows. max H (1) mn Whee s a maxmum sngula value of the fequency esponse functon matx, max s a mn mnmum sngula value of the fequency esponse functon matx. The condton numbe H of the FRF and SFRF ae shown n Fg.1. Fg.1 ndcates that the condton numbe value of the SFRF smalle than the condton numbe value of the FRF n the case of Case A. Theefoe, the nput dentfcaton usng the SFRF s supeo to dentfcaton usng the FRF n Case A. 1 3 FRF SFRF 1 3 FRF SFRF (a) Case A(All Fequency) (b) Case A(Fequency Zoomed) FRF SFRF (c) Case B Fg.1 Condton umbe 4. Hybd Input Identfcaton Ths pape nvestgates a new foce dentfcaton method by usng both stan measuement and acceleaton measuement. In ths pape, Ths new foce dentfcaton method s called hybd nput dentfcaton. In the hybd nput dentfcaton, stan s measued nea the exctaton pont, and acceleatons ae measued at othe ponts. It s abbevated as Hmx n ths pape. The numecal smulaton s conducted n the same manne as n Chapte 3. Locaton of the esponse ponts and exctaton ponts ae summazed n Table. In Case A-1,the exctaton foce s dentfed usng the FRF o the SFRF, n Case A-,the exctaton foce s dentfed by the Hmx. The dentfed and measued foce powe specta usng matx nveson method (M.I.) ae shown n Fg.11. The dentfed and measued foce powe specta usng appaent-mass matx method (A.M.) ae shown n Fg.1. The fequency aveaged eo s shown n Fg.13. Page 6 of 8 Inte-nose 14

7 Fequency Aveaged Eo[-] Powe Spectum[ ] Powe Spectum[ ] Inte-nose 14 Page 7 of 8 These esults ndcate that the nput dentfcaton usng Hmx has a hghe dentfcaton accuacy than the conventonal method (FRF). In addton, the nput dentfcaton accuacy usng Hmx s almost the same as the nput dentfcaton accuacy usng SFRF. Ths s an effectve method fom a pactcal pont of vew. Table Response and Exctaton Condtons 1 - M.I. FRF M.I. SFRF M.I. MIX Tue Fg.11 Foce Identfcaton(M.I.) Case Response Pont[m] A-1 A- Acceleaton Stan Acceleaton Stan Exctaton Pont[m] f1 f A.M. FRF A.M. SFRF A.M. Hmx Tue 1 - Fg.1 Foce Identfcaton(A.M.) FRF SFRF Hmx M.I. A.M. Fg.13 Fequency Aveaged Eo 5. COCLUSIOS Fo dvng pont FRF, fequency esponse functons ae affected by hghe-ode modes. Especally, the stan esponse (SFRF) s moe affected by hghe-ode modes than acceleaton esponse (FRF). The nput dentfcaton accuacy s mpoved by adoptng the dvng pont esponse. Especally, the nput dentfcaton usng stan esponse (SFRF) gves moe accuate esults than the acceleaton esponse (FRF). Ths pape nvestgated a hybd nput dentfcaton method: both acceleaton and stan ae used fo nput ndentfcaton; stan ae measued fo dvng pont esponse, and acceleatons ae measued fo othe esponses. Ths method has a hghe dentfcaton accuacy than the conventonal method. The dentfcaton accuacy of the hybd nput dentfcaton method s almost the same as the nput dentfcaton accuacy usng stan only esponse (SFRF).Ths s an effectve method fom a pactcal pont of vew. The nput dentfcaton accuacy of the appaent-mass matx method (A.M.)s much hghe than that of the matx nveson method(m.i.). Inte-nose 14 Page 7 of 8

8 Page 8 of 8 Inte-nose 14 REFERECES 1. Tou akagawa,yosho Koyanag, Expemental data analyss by the least squaes method, Unvesty of Tokyo Pess,198. Shgeyuk Kobayas,Takuya Yoshmua, Vaance Estmaton fo the Identfed Foce Specta Based on Appaent-Mass Matx, Tansactons of the Japan Socety of Mechancal Engnees, Sees C,Vol.76,o B.Hllay,D.J.Ewns, The Use of Stan Gauges n Foce Detemnaton and Fequency Response Functon Measuements,IMAC,(1984),pp Thomas Edwn Slaughte Kel, Assessment of Fequency Doman Foce Identfcaton Pocedues, Unvesty of Petoa,, 5. G. Thomas Rockln, John Cowley, Havad Vold, A Compason Of H1, H and Hv Fequency Response Functons, Poceedngs of the 3d Intenatonal Modal Analyss Confeence, (1985), pp.7-78 Page 8 of 8 Inte-nose 14