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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2010, Article ID 696512, 5 pages doi:10.1155/2010/696512 Research Article Health Monitoring for a Structure Using Its Nonstationary Vibration 1 1 1 2 Yoshimutsu Hirata, Mikio Tohyama, Mitsuo Matsumoto, and Satoru Gotoh SV Research Associates, 23-4-304, Yuigahama 2-chome, Kamakura-shi, Kanagawa 248-0014, Japan Waseda University, 1-104 Totsukama, Shinjuku-ku, Tokyo 169-8050, Japan Correspondence should be addressed to Mitsuo Matsumoto, matsu desk@jcom.home.ne.jp Received 26 May 2010; Accepted 4 August 2010 Academic Editor: K. M. Liew Copyright © 2010 Yoshimutsu Hirata et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The frequency distribution of a short interval period, the SIP distribution, obtained from the vibration of a structure that is excited by the force of non-stationary vibration is available for the robust estimation of the dynamic property of the structure. This paper shows experiments of the health monitoring of a model structure using the SIP distribution. Comparisons of SIP distributions with average DFT spectra are also shown. 1. Introduction 2. SIP Distributions One of the major problems after an earthquake, which causes The discrete Fourier transform (DFT) is used in many certain changes in the dynamic property of a structure, is the disciplines to obtain the spectrum or frequency of a signal. investigation of structural damage. In principle, we can check The DFT produces reasonable results for a large class of the change using a shaker to obtain the frequency response signal processes. However, we do not use the DFT for of astructure.However,itisdiﬃcult and not practical to detecting a short interval period because of its inherent shake a big structure before and after an earthquake to detect limitation of frequency resolution [2]. the change. Regardless of size or weight, all structures such The dominant frequency or period of a short-interval as buildings, towers, and bridges are vibrating due to the sequence W (m)(m = 0, 1,... , M) can be given by the natural force of winds, ground motions, or both. Informa- nonharmonic Fourier analysis [3]. In the process of the tion of the dynamic property of a building, for example, is analysis, we put W (x) = W (m), where x = m − M/2, and comprised of the forced vibration of a structure. Changes obtain Fourier coeﬃcients a( f )and b( f ) for an arbitrary of dynamic property reﬂect structural changes. Then, the frequency such that problem is how to extract information from the vibration of structures. It was shown by one of the authors that the frequency distribution of a short-interval period, the SIP distribution, M/2 W (x) sin 2πf x x=−M/2 of the forced nonstationary vibration corresponds to the a f = , M/2 2 sin 2πf x frequency response of a structure [1]. In this paper, exper- x=−M/2 iments of health monitoring of a model structure using its (1) nonstationary vibrations are shown, where SIP distributions M/2 W (x) cos 2πf x x=−M/2 are used for detecting the changes of the dynamic property b f = . M/2 cos 2πf x of a structure. x=−M/2 2 Advances in Acoustics and Vibration 1 1 0 0 100 1500 100 1500 Frequency (Hz) Frequency (Hz) (a) (b) Figure 1: Power frequency responses of model structures A (a) and B (b). 1 1 0 0 100 1500 100 1500 Frequency (Hz) Frequency (Hz) (a) (b) Figure 2: Examples of the clutter average spectrum of nonstationary vibration. 1 1 0 0 10 150 10 150 n n (a) (b) 1 1 0 0 10 150 10 150 n n (c) (d) 1 1 0 0 10 150 10 150 n n (e) (f ) Figure 3: Power frequency response of the model structure A estimated by the average DFT spectrum (a), (b), and (c) and the SIP distribution (b), (d), and (f ). (a) and (b) Stationary random vibration. (c)–(f ) Nonstationary vibration. Relative frequency is given by n = f /10. S( f ) S ( f ) Relative magnitude Relative magnitude Relative magnitude S ( f ) Relative magnitude Relative magnitude Relative magnitude B S( f ) Advances in Acoustics and Vibration 3 1 1 0 0 10 150 10 150 n n (a) (b) 1 1 0 0 10 150 10 150 n n (c) (d) 1 1 0 0 10 150 10 150 n n (e) (f ) Figure 4: Power frequency response of the model structure B estimated by the average DFT spectrum (a), (c), and (e) and the SIP disctribution (b), (d), and (e). (a) and (b) Stationary random vibration. (c)–(f ) Nonstationary vibration. Relative frequency is given by n = f /10. Hence, if we put where Δf< 1/M,wehave, from [1], S f − S f j i y x, f = a f sin 2πf x + b f cos 2πf x , D f − D f = Q (r ) ,(5) j i ij S f + S f j i M/2 (2) Y f = y x, f , where 0 <Q (r ) < 1and S( f ) is the power frequency x=−M/2 ij j response of a structure at a frequency f and so on. Hence, we get an approximation we have the dominant frequency f which satisﬁes S f ≈ kD f,(6) n n Y f = the maximum of Y f . (3) where k is an appropriate constant. It should be mentioned that the spectral resolution given by D( f ) depends little on It should be noted that we attain a least squares ﬁt of W (x) the length of the short sequence when Mf >1. to a sinusoid by y(x, f ). The SIP distribution is given by a number of dom- 3. Experiment inant frequencies (or periods) of short sequences which are fractions of measured data. Thus, the normalized SIP To conﬁrm the theoretical result shown by (6) and apply distribution D( f ) gives the probability of the dominant the SIP distribution to the health monitoring of a structure, frequency f foundinmeasureddata. experiments were made using a model structure. A model If we assume that a structure is excited by the force of structure is a wooden framework (18(W)×22(D)×27(H)cm) random noise and assign the frequency f in (1) such that which simulates a three-storied building with four struts. The strength of this model structure (the model structure f = f = nΔ f , n = 1, 2,... , N,(4) A) is changed by altering the struts, that is, the cross-section Relative magnitude Relative magnitude Relative magnitude Relative magnitude Relative magnitude Relative magnitude 4 Advances in Acoustics and Vibration DFT spectrum SIP distribution 10 150 10 150 n n (a) (b) DFT spectrum SIP distribution 10 150 10 150 n n (c) (d) Figure 5: Illustration of the health-monitoring of a structure by the DFT spectrum and the SIP distribution for random noise excitation (a), (b) and nonstationary noise excitation (c), (d). area of struts has a 25-percent decrease. This changed model spectra and the SIP distribution given by 2,400 dominant structure (a model structure B) is assumed to be the damaged frequency samples. Both estimations are much the same structure of the model A. when excited by the stationary random vibration, see Figures The frequency responses of the model structures A and 3(a) and 3(b).Diﬀerences arise in the case of nonstationary B, which are diﬃcult to measure in real cases, are shown vibrations, see Figures 3(c)–3(f ). in Figure 1, where the power frequency response S ( f )and Similarly, Figure 4 shows estimations of the power fre- S ( f ) are shown within the observation band. quency response of the model structure B. We see that the The force of stationary random vibration as well as SIP distribution is stable comparing with the DFT spectrum. nonstationary one was applied to excite a model structure, Figure 5 shows the illustration of the health-monitoring and the acceleration of the structure was measured at the of a structure using its nonstationary vibration, where the ﬁxed point of the frame. The measured data were provided model structure A changes gradually from time 10 and for the DFT analysis and SIP distribution. comes to the structure B at time 20. The DFT spectrum We assume that the average spectrum of nonstationary Figure 5(a) and the SIP distribution Figure 5(b) show the vibration is a clutter one which varies slowly with time, same change for random noise excitation. The diﬀerence so that we represent a nonstationary vibration as a set of the both method arises when the structure is excited by of random noise sequences each having a diﬀerent clutter nonstationary noises Figures 5(c) and 5(d). It seems that the average spectrum; see Figure 2. SIP distribution is stable comparing with the DFT spectrum. The measured data sequence was divided into 2,400 short-interval sequences for the nonharmonic Fourier anal- ysis to get a SIP distribution, that is, the dominant frequency 4. Conclusions of each sequence is given by the analysis of 60 sampled data. To compare the SIP distribution with the average of DFT A number of buildings, bridges, and towers have been spectra, the measured data sequence was also divided into constructed in past decades. Consequently, there are many 240 sequences, so that the DFT frequency, f ,isgiven by decrepit structures which need to be reconstructed. One may nF/600, where F is a sampling frequency. Corresponding to remember the accident in Minneapolis, USA where thirteen the DFT frequency, we put the frequency of the nonharmonic persons were killed by the sudden collapse of an old bridge Fourier analysis (see (1)) such that f = f . over the Mississippi. The health monitoring of structures is Figure 3 shows the power frequency response of the an important mean for security against such an accident. The model structure A estimated by the average of 240 DFT method described in this paper might be available. Further Time Time Time Time Advances in Acoustics and Vibration 5 experiments using a real bridge, for example, will conﬁrm the proposed method, though it will take decades. References [1] Y. Hirata, “Estimation of the frequency response of a structure using its non-stationary vibration,” Journal of Sound and Vibration, vol. 313, no. 3–5, pp. 363–366, 2008. [2] S. M. Kay and S. L. Marple Jr., “Spectrum analysis—a modern perspective,” Proceedings of the IEEE, vol. 69, no. 11, pp. 1380– 1419, 1981. [3] Y. Hirata, “Non-harmonic Fourier analysis available for detect- ing very low-frequency components,” Journal of Sound and Vibration, vol. 287, no. 3, pp. 611–613, 2005. 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Advances in Acoustics and Vibration – Hindawi Publishing Corporation
Published: Sep 1, 2010
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