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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2008, Article ID 261758, 8 pages doi:10.1155/2008/261758 Research Article Structural Condition Monitoring by Cumulative Harmonic Analysis of Random Vibration 1 2 3 Yoshinori Takahashi, Toru Taniguchi, and Mikio Tohyama Faculty of informatics, Kogakuin University, 1-24-2 Nishi-shinjyuku Shinjyuku-ku, Tokyo 163-8677, Japan Information Technology Research Organization, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan Waseda University, Global Information and Telecommunication Institute, 1011 Okuboyama, Nishi-Tomida Honjo-shi, Saitama 367-0035, Japan Correspondence should be addressed to Yoshinori Takahashi, firstname.lastname@example.org Received 11 October 2007; Revised 21 January 2008; Accepted 3 June 2008 Recommended by Lars Hakansson Analysis of signals based on spectral accumulation has great potential for enabling the condition of structures excited by natural forces to be monitored using random vibration records. This article describes cumulative harmonic analysis (CHA) that was achieved by introducing a spectral accumulation function into Berman and Fincham’s conventional cumulative analysis, thus enabling potential new areas in cumulative analysis to be explored. CHA eﬀectively enables system damping and modal overlap conditions to be visualized without the need for transient-vibration records. The damping and modal overlap conditions lead to a spectral distribution around dominant spectral peaks due to structural resonance. This distribution can be revealed and emphasized by CHA records of magnitude observed even within short intervals in stationary random vibration samples. Copyright © 2008 Yoshinori Takahashi 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. 1. INTRODUCTION of the dominant components. As variance in the distribution increases, the modal bandwidth widens, thus increasing This paper describes spectral changes visualized in a struc- damping. Hirata  also developed a method of spectral tural-vibration system using cumulative harmonic analysis analysis, called nonharmonic Fourier analysis, for extracting the dominant low-frequency components of a vibration (CHA)  under random vibration conditions. Spectral changes in vibrations should be informative for monitoring record in an SIP. However, spectral analysis in SIPs is and diagnosing system health in structures. However, it normally diﬃcult for frequency resolution and artifacts, because of time-windowing functions . is diﬃcult to track variations in the structural-transfer functions independent of the source-signal characteristics. Resonance can be interpreted as a spectral-accumulation This is because the transfer-function analysis of structural- process of in-phase sinusoidal components with a ﬁxed- vibration systems generally requires the speciﬁcations of the phase lag. Therefore, as the impulse response for a resonant excitation source [2, 3]. system is written as an inﬁnite series from the signal- Hirata  proposed a method of monitoring invisible processing viewpoint, the eﬀective length of the sequence changes in structures based on the frequency distributions increases as system damping decreases but this is short of the dominant spectral components in every short interval under heavy damping conditions. We examined the spectral- accumulation process on a sample-by-sample basis to visu- period (SIP) under random and nonstationary vibration conditions without any speciﬁc source-signal assumptions or alize the spectral changes in the dominant modal-resonant requirements. Changes in the modal resonance bandwidth, components under random vibration. which represent the damping conditions for modal vibration Berman and Fincham  previously formulated cumu- and must therefore be a signiﬁcant indicator of the health of lative spectral analysis (CSA) using a stepwise time- a structure, can be evaluated from the frequency histogram window function to evaluate the transient characteristics of 2 Advances in Acoustics and Vibration Signal x(n) N = 16 ··· Where −1 −jΩ z = e 02 4 6 8 10 12 14 (n) −j 2πk/N = e n = 0 DFT 02 4 6 8 10 12 14 (m) n = 1 DFT DFT 02 4 6 8 10 12 14 (m) n = 2 DFT DFT DFT 02 4 6 8 10 12 14 (m) DFT N -point DFT with zero padding Figure 1: Schematic for cumulative harmonic analysis (CHA). loudspeaker systems. We looked at the cumulative properties into spectral accumulation function w(n), we have of a spectral display of vibration records including transfer function changes. In Section 2, we introduce a forgetting function into CSA instead of using a stepwise time window. −1 −m CHA(n, z ) = (m +1)x(m)z We called this cumulative harmonic analysis (CHA), and m=0 its purpose was to emphasize the spectral-accumulation (4) −0 −1 = 1x(0)z +2x(1)z process, including the increasing resonant spectral peak, on a sample-by-sample basis . We describe CHA’s eﬀectiveness −2 −n +3x(2)z +··· + nx(n)z in enabling changes in the transfer functions to be visualized through numerical simulation experiments on a single vibrating system in Section 3.In Section 4, we also describe in the z-plane including the unit circle where the Fourier CHA’s eﬀectiveness in a two-degrees-of-freedom (2-DOF) transform can be deﬁned. Figure 1 has a schematic of CHA. vibrating system. Section 5 discusses CHA visualization We use a triangular window as the spectral accumulation under conditions with damping changes. function, w(m) = m + 1, in this paper where the math- ematical expression in the accumulation eﬀect is simple. 2. FORMULATION OF CUMULATIVE HARMONIC However, we can freely deﬁne accumulation function w(n) αm ANALYSIS (CHA) FROM CSA as an exponential function, w(m) = e , based on the degree of necessary spectral accumulation (or forgetting eﬀect). We formulate CHA by introducing a forgetting function The eﬀect of the transfer-function pole on frequency into CSA that corresponds to the spectral-accumulation characteristics can be emphasized by CHA. Assume that we function. Assume that we have a signal sequence, x(n), and have a simple decaying sequence deﬁne a spectral-accumulation function, w(n). We deﬁne the cumulative harmonic analysis (CHA) of x(n)as x(n) ≡ a (n = 0, 1, 2,...), 0 < |a| < 1. (5) −jΩ −jΩ CHA(n, e ) ≡ X (m, e ), (1) m=0 If we take the limit for CHA of the sequence above as where −jΩ −jΩm X (m, e ) ≡ w(m)x(m)e ,0 ≤ m ≤ n. (2) −1 m −m lim CHA(n, z ) ≡ lim (m +1)a z = , Substituting a forgetting function such as 2 −1 n→∞ n→∞ (1 − az ) m=0 w(m) ≡ m +1, 0 ≤ m ≤ n,(3) (6) Samples (n) −1 CHA(0, z ) −1 CHA(1, z ) −1 CHA(2, z ) −1 CHA(3, z ) Frequency bin (k) Yoshinori Takahashi et al. 3 Impulse response −1 2 4 6 8 10 121416 18 20 Sampled time (n) (a) Normalized CHA magnitude −1 −2 −2 16 18 10 12 14 6 8 Sampled time (n) (b) Normalized CSA magnitude −1 −2 −2 14 16 18 10 12 6 8 Sampled time (n) (c) Im Re −11 −j jΩ a =|a|e |a|= 0.8 Ω =±π/4 (d) Figure 2: Samples of CHA (panel b) and CSA (panel c) magnitude displays for impulse response of single-degree-of-freedom resonance system where poles of transfer function are located at π/4in z-plane. Panel (a) illustrates sequence of impulse responses. Maximum magnitude is normalized to unity at every instant in both panels (b) and (c) and we set N = 800. then we can see that CHA virtually increases the order of Consequently, the modal resonance is visualized as a the pole compared with regular discrete Fourier transform way for the dominant-frequency components of vibration (DFT) (w(n) = 1) records to be narrowed down to dominant elements. Changes in the resonance of the structural transfer functions can be expected to be visualized even under stationary −1 m −m lim DFT(n, z ) = lim a z = . (7) −1 n→∞ n→∞ 1 − az random-vibration conditions. Figure 2 shows what eﬀect m=0 Normalized angular frequency (π ) Normalized angular frequency (π ) 4 Advances in Acoustics and Vibration CHA magnitude 0.75 0.5 (a) 0.25 0 0 STFT 0.75 (b) 0.5 0.25 MA spectrum 0.75 (c) 0.5 0.25 −30 (d) (e) 00.10.20.30.126.96.36.199.80.91 Accumulation time (normalized by impulse-response length) z-plane 0 1 −5 −5 0 −10 −10 −1 0.20.25 0.3 0.20.25 0.3 −10 1 Normalized frequency (π ) Normalized frequency (π ) Re jΩ z = ae a = 0.98 Ω = π/4 (g) (f ) (h) Figure 3: Examples of CHA and spectrograms for random record of single-degree-of-freedom (SDOF) system. (a) CHA magnitude and (b) spectrogram using STFT conditions are frame length, that is, 0.18 with zero padding under conditions where total Fourier-transform length is 91 and frame hop size is 1/10 frame length. Time intervals are normalized by impulse response record length (1000 points). (c) Moving averaged spectrogram using averaging STFT, (b) frame-by-frame, (d) random record to be analyzed, and (e) impulse response record of vibration system where length is deﬁned by reverberation time. (f ) Magnitude response and normalized frequency, (g) close up of CHA magnitude in (a) at ﬁnal observation instant accumulation time of 1, and (h) pole plot of transfer function for impulse response in (e). CHA has against regular DFT using real and causal sequences process is emphasized by CHA where the resonant peak is including complex conjugate poles. In this example, pole increasing. a in (5) has been expanded to a complex number for a more general demonstration. The locations of the complex- 3. CHA EXAMPLE OF RANDOM VIBRATION RECORD IN conjugate poles are shown in panel (d). The impulse SINGLE-DEGREE-OF-FREEDOM (SDOF) SYSTEM response is illustrated in panel (a), the magnitude record for CSA is in panel (c), and that for CHA is in (b). The maximum Assume that we have a random vibration record observed in magnitude is normalized to unity at every instant in both a single-degree-of-freedom (SDOF) system. These types of panels (b) and (c). We can see the spectral-accumulation structural-vibration samples are normally easy to obtain in Magnitude (dB) Normalized frequency (π ) Magnitude (dB) Im (dB) Yoshinori Takahashi et al. 5 n n n ··· ··· DFT DFT DFT DFT DFT DFT DFT DFT DFT ··· 1/2 1/3 1/2 2/3 CHA gram STFT gram MA gram (a) (b) (c) Figure 4: Algorithms for (a) CHA, (b) STFT, and (c) MA. DFT means an N -point DFT with zero padding. 0 0.8 AB < 0 ΔΩ −5 0.7 p −10 A B −1 H (z ) = + −1 −1 0.6 (1 − z z ) (1 − z z ) p p 1 2 z-plane −15 0.22 0.24 0.26 0.28 0.3 0.60.70.8 Normalized frequency (π ) Re jΩ z = ae ΔΩ = 3h Ω p 0 p a = 0.98 h = 0.04 Ω = π/4 (a) 0 0.8 AB > 0 −5 0.7 −10 0.6 z-plane −15 0.22 0.24 0.26 0.28 0.3 0.60.70.8 Normalized frequency (π ) Re (b) Figure 5: Pole/zero plots for two-degrees-of-freedom systems (2-DOF systems) and magnitude responses. SIPs due to the natural force of winds, ground movement, or Here, we deﬁne N as the length of an impulse response both  without speciﬁcally having to prepare source signals, record, and T denotes the sampling period of discrete which is impractical for large structures. sequences. Figure 3 shows examples of CHA random-vibration In Figure 3, the CHA, STFT, and MA spectra with records, including an impulse response. In the examples, we the peaks plots were measured on a sample-by-sample or have deﬁned the damping factor, h,as frame-by-frame basis and the maximum magnitude records were normalized to unity in every instance of observation. − ln a We can see that CHA in panel (a) visualizes the resonant ≡ 0.078. (8) h = properties of the SDOF system better than the conventional spectrogram using STFT in panel (b). The spectrograms The axis representing time in Figure 2 is normalized by the using STFT conditions are the frame length, that is, 0.18 length of the impulse response record, which is given by the with zero padding under conditions where the total Fourier- reverberation time, T , which is estimated as transform length is 91 and the frame hop size is 1/10 the frame length (the time intervals are normalized by 6.9T T ≡ NT . (9) R s the impulse response record length (1000 points)). The − ln a Magnitude (dB) Magnitude (dB) Im Im 6 Advances in Acoustics and Vibration 0.4 0 0.4 0 CHA magnitude CHA magnitude 0.35 0.35 0.3 0.3 0.25 0.25 0.2 −20 0.2 −20 02 4 6 8 02 4 6 8 0.4 0 0.4 0 MA spectrum MA spectrum 0.35 0.35 0.3 0.3 0.25 0.25 0.2 −20 0.2 −20 02 4 6 8 02 4 6 8 0.4 0.4 Peak tracking Peak tracking 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 02 4 6 8 02 4 6 8 Accumulation time (normalized by impulse-response length) Accumulation time (normalized by impulse-response length) CHA CHA MA MA (a) (b) Figure 6: CHA and MA displays and tracking of prominent peaks under 2-DOF-system conditions. Panel (a): 2-DOF system without zeros, panel (b): 2-DOF system with zero. MA spectrum in panel (c) also seems to enable stable of damping on the spectral distribution. Structural-vibrating analysis of resonant properties. The MA spectrogram uses systems at low frequencies generally have low modal overlap. averaging STFT frame-by-frame according to the frame However, if there is a pair of adjacent poles, the damping progression. Panel (g) shows the CHA peak corresponding conditions change the modal overlap, which is deﬁned as to the impulse response spectrum peak in panel (f ). This [9, 10] means that CHA can promptly and eﬀectively be used to determine the presumed modal frequency of the transfer ∼ M ≡ , B πδ = πhω , (10) = p Δω function included in the noise signal. The accumulation time required to obtain a stable CHA-magnitude record seems where B is the modal bandwidth, Δω is the average modal to be around one-half the length of the impulse-response spacing, and ω is the modal angular frequency of interest. record. The spectral components of CHA obtained in every Figure 5 shows examples of the pole/zero plots for a 2-DOF instance are interpolated by DFT with zero padding after system under low modal-overlap conditions. Panel (b) is a the vibration record. Accumulating the interpolated spectra plot with a single zero, and panel (a) is that without zeros provides an accurate estimate of the modal frequency and between the poles. A pair of same sign residues yields a zero does not suﬀer from the artifacts of the time-windowing between the poles, and a pair of poles with diﬀerent sign function; consequently, the modal properties are emphasized residues has no zeros [11–13]. Here, we set Δω as the distance by virtually increasing the degree of the pole. between the two pole angular frequencies in Figure 5. ωp is given by Ω . We varied the damping conditions to control 4. CHA MONITORING OF MODAL OVERLAP the modal overlap. CONDITIONS IN TWO-DEGREES-OF-FREEDOM Figure 6 shows the CHA magnitudes and MA spectrums (2-DOF) SYSTEM forthe 2-DOFsystemillustrated in Figure 5.Wecan seen that CHA in both panels (a) and (b) visualizes the resonant The damping condition is a signiﬁcant indicator of structural properties of the 2-DOF system better than the conventional damage. This section explains how CHA visualizes the eﬀect MA spectrograms. Panel (a) is a plot without zeros, and panel Normalized frequency (π ) (dB) (dB) Normalized frequency (π ) (dB) (dB) Yoshinori Takahashi et al. 7 0 0 0.25 0.25 −5 −5 0.28 0.28 −10 −10 0.20.30.4 0.20.30.4 Normalized frequency (π ) Normalized frequency (π ) CHA CHA p peaks eaks hist histog ogr ra am m 0.4 0.4 0.3 0.3 0.25 0.25 0.2 0.2 0.28 0.28 0.1 0.1 0 0 0.20.30.4 0.20.30.4 Normalized frequency (π ) Normalized frequency (π ) MA peaks histogram 0.4 0.4 0.3 0.3 0.26 0.25 0.27 0.34 0.2 0.2 0.34 0.1 0.1 0 0 0.20.30.4 0.20.30.4 Normalized frequency (π ) Normalized frequency (π ) (a) (b) Figure 7: CHA and MA peak histogram for 2-DOF system. Panel (a): 2-DOF system without zeros, panel (b): 2-DOF system with zero. (b) is one with a single zero. The modal overlap conditions modal-overlap conditions changing from M = 0.25 to M = 1 in both panels were set to M = 0.5. The top and middle during CHA monitoring with or without a zero between panels illustrate the CHA and MA records. The bottom the poles. Panel (a) shows the case without zeros, and panel panels in (a) or (b) show the results of spectral tracking  (b) includes a zero between the poles. The bottom panels for prominent peaks of CHA or MA spectrograms. Figure 7 illustrate the peak-tracking results of the CHA spectrogram. shows histograms of all the peaks in Figure 6. We can see that We can see that the variance in the peak-tracking lines of the the CHA peaks correspond to the modal frequencies. CHA peaks increases according to the change in damping in both panels. 5. CHA MONITORING UNDER CONDITIONS WITH DAMPING CHANGES 6. SUMMARY This section discusses how CHA visualizes spectral changes We assessed CHA and conﬁrmed through numerical simula- due to damping conditions. Figure 8 shows examples of tion experiments that it eﬀectively enabled system damping Magnitude (dB) Normalized number Normalized number Magnitude (dB) Normalized number Normalized number 8 Advances in Acoustics and Vibration Changing under stationary and nonstationary vibration conditions. M = 0.25 M = 1 0 0 modal However, substantial ﬁeld tests are required to develop a overlap −5 −5 more practical monitoring system. −10 −10 0.20.30.4 0.20.30.4 REFERENCES 0.4 CHA magnitude 0.35  Y. Takahashi, M. Tohyama, and Y. Yamasaki, “Cumulative 0.3 spectral analysis for transient decaying signals in a transmis- 0.25 sion system including a feedback loop,” Journalofthe Audio 0.2 Engineering Society, vol. 54, no. 7-8, pp. 620–629, 2006. −20  P. J. Halliday and K. Grosh, “Maximum likelihood estimation 0.4 of structural wave components from noisy data,” The Journal 0.35 of the Acoustical Society of America, vol. 111, no. 4, pp. 1709– 0.3 1717, 2002. 0.25  J. G. McDaniel and W. S. 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The results indicate that CHA could be an eﬃcient method of monitoring and diagnosing structures without signal-source requirements Normalized frequency (π ) Normalized frequency (π ) Magnitude(dB) Magnitude(dB) Magnitude(dB) Magnitude(dB) (dB) (dB) International Journal of Rotating Machinery International Journal of Journal of The Scientific Journal of Distributed Engineering World Journal Sensors Sensor Networks Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Volume 2014 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Submit your manuscripts at http://www.hindawi.com Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 VLSI Design Advances in OptoElectronics International Journal of Modelling & Aerospace International Journal of Simulation Navigation and in Engineering Engineering Observation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2010 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 International Journal of Active and Passive International Journal of Antennas and Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014
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