Advaces i Acoustics ad Vibratio Volume 2, Article ID 69652, 5 pages doi:.55/2/69652 Research Article Health Moitorig for a Structure Usig Its Nostatioary Vibratio Yoshimutsu Hirata, Mikio Tohyama, Mitsuo Matsumoto, ad Satoru Gotoh 2 SV Research Associates, 23-4-34, Yuigahama 2-chome, Kamakura-shi, Kaagawa 248-4, Japa 2 Waseda Uiversity, -4 Totsukama, Shijuku-ku, Tokyo 69-85, Japa Correspodece should be addressed to Mitsuo Matsumoto, matsu desk@jcom.home.e.jp Received 26 May 2; Accepted 4 August 2 Academic Editor: K. M. Liew Copyright 2 Yoshimutsu Hirata et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. The frequecy distributio of a short iterval period, the SIP distributio, obtaied from the vibratio of a structure that is excited by the force of o-statioary vibratio is available for the robust estimatio of the dyamic property of the structure. This paper shows experimets of the health moitorig of a model structure usig the SIP distributio. Comparisos of SIP distributios with average DFT spectra are also show.. Itroductio Oe of the major problems after a earthquake, which causes certai chages i the dyamic property of a structure, is the ivestigatio of structural damage. I priciple, we ca check the chage usig a shaker to obtai the frequecy respose ofastructure.however,itisdifficult ad ot practical to shake a big structure before ad after a earthquake to detect the chage. Regardless of size or weight, all structures such as buildigs, towers, ad bridges are vibratig due to the atural force of wids, groud motios, or both. Iformatio of the dyamic property of a buildig, for example, is comprised of the forced vibratio of a structure. Chages of dyamic property reflect structural chages. The, the problem is how to extract iformatio from the vibratio of structures. It was show by oe of the authors that the frequecy distributio of a short-iterval period, the SIP distributio, of the forced ostatioary vibratio correspods to the frequecy respose of a structure []. I this paper, experimets of health moitorig of a model structure usig its ostatioary vibratios are show, where SIP distributios are used for detectig the chages of the dyamic property of a structure. 2. SIP Distributios The discrete Fourier trasform (DFT) is used i may disciplies to obtai the spectrum or frequecy of a sigal. The DFT produces reasoable results for a large class of sigal processes. However, we do ot use the DFT for detectig a short iterval period because of its iheret limitatio of frequecy resolutio [2]. The domiat frequecy or period of a short-iterval sequece W(m)(m =,,..., M) ca be give by the oharmoic Fourier aalysis [3]. I the process of the aalysis, we put W(x) = W(m), where x = m M/2, ad obtai Fourier coefficiets a( f )adb( f ) for a arbitrary frequecy such that a ( f ) = b ( f ) = x= M/2 W(x) si ( 2πfx ) x= M/2 si 2( 2πfx ), x= M/2 W(x) cos ( 2πfx ) x= M/2 cos 2( 2πfx ). ()
2 Advaces i Acoustics ad Vibratio SA ( f ) SB ( f ) 5 Frequecy (Hz) 5 Frequecy (Hz) Figure : Power frequecy resposes of model structures A ad B. S( f ) S( f ) 5 Frequecy (Hz) 5 Frequecy (Hz) Figure 2: Examples of the clutter average spectrum of ostatioary vibratio. Relative magitude Relative magitude Relative magitude 5 5 (c) 5 (e) Relative magitude Relative magitude Relative magitude 5 5 (d) 5 (f) Figure 3: Power frequecy respose of the model structure A estimated by the average DFT spectrum,, ad (c) ad the SIP distributio, (d), ad (f). ad Statioary radom vibratio. (c) (f) Nostatioary vibratio. Relative frequecy is give by = f /.
Advaces i Acoustics ad Vibratio 3 Relative magitude Relative magitude Relative magitude 5 5 (c) 5 (e) Relative magitude Relative magitude Relative magitude 5 5 (d) 5 (f) Figure 4: Power frequecy respose of the model structure B estimated by the average DFT spectrum, (c), ad (e) ad the SIP disctributio, (d), ad (e). ad Statioary radom vibratio. (c) (f) Nostatioary vibratio. Relative frequecy is give by = f /. Hece, if we put y ( x, f ) = a ( f ) si ( 2πfx ) + b ( f ) cos ( 2πfx ), Y ( f ) = M/2 x= M/2 y 2( x, f ), we have the domiat frequecy f p which satisfies (2) Y ( f p ) = the maximum of Y ( f ). (3) It should be oted that we attai a least squares fit of W(x) to a siusoid by y(x, f p ). The SIP distributio is give by a umber of domiat frequecies (or periods) of short sequeces which are fractios of measured data. Thus, the ormalized SIP distributio D( f p ) gives the probability of the domiat frequecy f p foudimeasureddata. If we assume that a structure is excited by the force of radom oise ad assig the frequecy f i () such that f = f = Δ f, =, 2,..., N, (4) where Δ f</m, wehave,from[], D ( ) ) f j D ( ) f i = Qij (r) S( f j S ( ) f i S ( ) ( ), (5) f j + S fi where < Q ij (r) < ads( f j ) is the power frequecy respose of a structure at a frequecy f j ad so o. Hece, we get a approximatio S ( f ) kd ( f ), (6) where k is a appropriate costat. It should be metioed that the spectral resolutio give by D( f ) depeds little o the legth of the short sequece whe Mf >. 3. Experimet To cofirm the theoretical result show by (6) ad apply the SIP distributio to the health moitorig of a structure, experimets were made usig a model structure. A model structure is a woode framework (8(W) 22(D) 27(H)cm) which simulates a three-storied buildig with four struts. The stregth of this model structure (the model structure A) is chaged by alterig the struts, that is, the cross-sectio
4 Advaces i Acoustics ad Vibratio DFT spectrum SIP distributio 2 5 DFT spectrum 2 5 SIP distributio 2 5 (c) 2 5 (d) Figure 5: Illustratio of the health-moitorig of a structure by the DFT spectrum ad the SIP distributio for radom oise excitatio, ad ostatioary oise excitatio (c), (d). area of struts has a 25-percet decrease. This chaged model structure (a model structure B) is assumed to be the damaged structure of the model A. The frequecy resposes of the model structures A ad B, which are difficult to measure i real cases, are show i Figure, where the power frequecy respose S A ( f )ad S B ( f ) are show withi the observatio bad. The force of statioary radom vibratio as well as ostatioary oe was applied to excite a model structure, ad the acceleratio of the structure was measured at the fixed poit of the frame. The measured data were provided for the DFT aalysis ad SIP distributio. We assume that the average spectrum of ostatioary vibratio is a clutter oe which varies slowly with time, so that we represet a ostatioary vibratio as a set of radom oise sequeces each havig a differet clutter average spectrum; see Figure 2. The measured data sequece was divided ito 2,4 short-iterval sequeces for the oharmoic Fourier aalysis to get a SIP distributio, that is, the domiat frequecy of each sequece is give by the aalysis of 6 sampled data. To compare the SIP distributio with the average of DFT spectra, the measured data sequece was also divided ito 24 sequeces, so that the DFT frequecy, f,isgiveby F/6, where F is a samplig frequecy. Correspodig to the DFT frequecy, we put the frequecy of the oharmoic Fourier aalysis (see ()) such that f = f. Figure 3 shows the power frequecy respose of the model structure A estimated by the average of 24 DFT spectra ad the SIP distributio give by 2,4 domiat frequecy samples. Both estimatios are much the same whe excited by the statioary radom vibratio, see Figures 3 ad 3. Differeces arise i the case of ostatioary vibratios, see Figures 3(c) 3(f). Similarly, Figure 4 shows estimatios of the power frequecy respose of the model structure B. We see that the SIP distributio is stable comparig with the DFT spectrum. Figure 5 shows the illustratio of the health-moitorig of a structure usig its ostatioary vibratio, where the model structure A chages gradually from time ad comes to the structure B at time 2. The DFT spectrum Figure 5 ad the SIP distributio Figure 5 show the same chage for radom oise excitatio. The differece of the both method arises whe the structure is excited by ostatioary oises Figures 5(c) ad 5(d). It seems that the SIP distributio is stable comparig with the DFT spectrum. 4. Coclusios A umber of buildigs, bridges, ad towers have bee costructed i past decades. Cosequetly, there are may decrepit structures which eed to be recostructed. Oe may remember the accidet i Mieapolis, USA where thirtee persos were killed by the sudde collapse of a old bridge over the Mississippi. The health moitorig of structures is a importat mea for security agaist such a accidet. The method described i this paper might be available. Further
Advaces i Acoustics ad Vibratio 5 experimets usig a real bridge, for example, will cofirm the proposed method, though it will take decades. Refereces [] Y. Hirata, Estimatio of the frequecy respose of a structure usig its o-statioary vibratio, Joural of Soud ad Vibratio, vol. 33, o. 3 5, pp. 363 366, 28. [2] S. M. Kay ad S. L. Marple Jr., Spectrum aalysis a moder perspective, Proceedigs of the IEEE, vol. 69, o., pp. 38 49, 98. [3] Y. Hirata, No-harmoic Fourier aalysis available for detectig very low-frequecy compoets, Joural of Soud ad Vibratio, vol. 287, o. 3, pp. 6 63, 25.
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