|Abstract: ||時頻分析(time-frequency analysis)係一個近期發展的數值分析技術，可展現一個訊號在時間域與頻率域上的能量分布情形，適合觀察物體隨時間變化之動態特性。希爾伯特-黃轉換(Hilbert-Huang Transform, 簡稱為HHT)係眾時頻分析方法之一，因具”局部隨適性”與”後待定基底函數”兩特性，所以常被引用在非線性(Nonlinear)且非穩態(Nonstationary)訊號分析上，且特性模態函數(Intrinsic Mode Function，簡稱為IMF)更實現了瞬時頻率(Instantaneous Frequency，簡稱為IF)的計算方法與其物理意義。|
希爾伯特轉換(Hilbert Transform)係將訊號視為一串振幅隨時間調變 (Amplitude Temporal Modulated Function)函數與頻率隨時間調變 (Frequency Temporal Modulated Function)函數乘積之數列組合，與傅立葉轉換(Fourier Transform)比較，傅立葉將訊號視為一串不隨時變的定振幅函數與定頻率函數乘積之數列組合，希爾伯特轉換在定義上就符合非穩態系統(Non-stationary System)的基本要求。經改良成希爾伯特-黃轉換後，其”後待定基底”(posteriori base)的特性，解除了相關數值轉換以正弦波或其他函數當成”預設基底”函數的先天限制，讓基底函數的本質與訊號分解過程無關，隨適性基底(adaptive base)的觀念更在訊號處理領域引發討論，不預設基底函數而讓資料自身產生基底函數，在通訊領域之應用雖較傅立葉或小波轉換沒效率，但很適合運用在地球科學相關科學領域。
地震波係由震源板塊破裂放出能量，經波行路徑上板塊與岩基地盤介質傳播後傳至地表，地震學上已解析出地震波形由數個不同的波相(Phase)所組成，係為一種非穩態訊號。又當震央放出巨大能量時，介質物質產生非線性破壞，強震波形應為一具非線性特性之非穩態訊號。本文係以HHT來分析結構體強震資料，以HHT在時頻域之解析力，探討結構體受震後的結構安全診斷(Structural Health Monitoring, 簡稱為SHM)相關資訊與參數，並借助IF來解讀結構體受振後的細微變化。
一個多自由度(MDOF)結構體受水平向力之理論解(analytical solution)為數個結構模態(mode)的線性組合，此理論解是基於力學與數學算得，且結構須於線性穩態的條件下。當我們把真實大樓(real building)頂部測得之強震加速度歷時(strong-motion waveform)資料直接展現於時頻譜上時，可以觀察到譜上之能量會集中於某些固定頻率值，該值可視為該結構模態之主頻，譜上還可以看到模態主頻值隨時間有些跳動。這些結構模態行為(modal behavior)並非由理論解算出，且時間軸上的小跳動特性與結構健康狀況有關。藉著HHT在時頻域之精準數值展現，真實大樓受震結構模態行為可直接由判讀時頻譜獲得。
除了直接判讀時頻譜之外，還有一些數值計算方式與技術可以幫助我們獲得結構安全診斷SHM相關資訊與參數，包括訊號強化技術(signal enhancement skills)、本分析建立之時頻域放大函數(time-frequency domain amplification function, 簡稱為T.F.AF)、模態時間曲線(modal temporal variation curve, 簡稱為MTVC)的萃取技術與瞬時頻率在SHM的應用等。在文中，我們發展了一套新方法，整合了所有上述提到的計算技術，從結構受震反應相關時頻譜上找出各種SHM相關資訊與參數，將該法稱為 HHT SHM method。文中並引用了振動台(shaking-table)實驗資料與真實大樓之強震觀測資料來完成"方法驗證"與"方法展現"兩件重要工作。
關鍵字：時頻分析、希爾伯特-黃轉換法、結構物強震分析、基振分析、結構安全診斷、雙站法、時頻域放大函數、模態時間曲線;The time-frequency analysis is a brand-new new numerical technique. The energy distribution pattern of a signal can be demonstrated simultaneously on the time domain and frequency domain, it′s a powerful tool for the observation of a time-variant system. The Hilbert-Huang Transform (HHT) is one of the time-frequency analysis methods. The good local adaption and the posteriori base are the unique specialities of the HHT. With these characteristics, HHT is found useful for the nonlinear and non-stationary signal analysis. Additionally, the definition of the Intrinsic Mode Function (IMF) makes the calculation of the instantaneous frequency (IF) possibly. The physical meaning of IF was also clearly demonstrated by the HHT.
The original signal is decomposed into the summation of the production of amplitude temporal modulated function and the frequency temporal modulated function by the HHT. Meanwhile, the Foruier decompose the signal into the summation of the production of constant amplitude function and the constant frequency function. The time-varriant characteristic of the Hilbert Transform makes the analysis for the non-stationary system possible. The Hilbert-Huang Transform, the improved Hilbert Transform, the speciality of the posteriori base makes the method totally adaptive. The decomposition procedures are not influenced by the base function anymore. The limitations and congenital defects coming from the base function (sinusoid or wavelet function) are totally disappeared. Contrary to almost all the previous methods, the HHT method is intuitive, direct, a posteriori, and adaptive, with the basis of the decomposition based on and derived from the data. The HHT might not be efficient for the artificial communication signal problems. It’s found suitable for the geophysical related analysis.
The earthquake energy was generated from hypocenter tectonic activities. The released energy passed through underground geographic media, reached on the specific site conditions, impacted the surface of the ground. Seismologically speaking, the earthquake waveform can be decomposed into several different wave phases. With several phases, the earthquake is not a stationary signal. Furthermore, for a strong earthquake, the destructive energy makes the underground geographic media into the nonlinear deformation stage. For a strong-motion earthquake record, it′s not only a non-stationary signal but also having nonlinear characteristics. In this article, we use HHT to analyze the strong-motion records of structures. The observations on the time-frequency spectrum are what we planned to extract detail dynamic properties of building. Structural Health Monitoring (SHM) parameters and information can be extracted from the HHT result directly. The IF is also introduced to explain the vivid difference of the SHM condition.
The theoretical solution of a Multiple-Degree-Of-Freedom (MDOF) structure excited by horizontal force is the combination of several individual modes that are supported by mechanical theorems and mathematical formulas. When we directly demonstrate the waveform from the building strong-motion response waveform on the time-frequency spectrum, the energy distribution usually concentrates at some frequencies, and also the temporal variations of each frequency band are clearly shown as well. Those temporal variations of the frequencies are the actual performances of structure during the vibration, even though those "modal behaviors" are not coming from the solution of the theoretical linear and stationary governing equations. By the vantages of the HHT, the modal behaviors of a real building can be obtained easily by directly reading the time-frequency spectrum of the acceleration response record.
In addition to read the response on the spectrum, some numerical steps are found even more helpful on collecting SHM information, including the signal enhancement skills, the newly defined time-frequency domain amplification function (T.F.AF) ,the extraction of modal temporal variation curve (MTVC), and the SHM application of the IF. We develop a new method called the HHT SHM method which integrated all the numerical steps mentioned above. The measurement of shaking-table experiment and the real building observation data are used to show the performance and the method-validation.
The procedures are described step by step as follows. First, the original nonlinear and nonstationary real building strong-motion response signal is firstly been transferred into the wave-propagating properties, the T.F.AF. The T.F.AF which is the core of the method can give dynamic parameters results through all the phases in an earthquake event. Then, after adopting selective and mechanical judgment, the useful structural modal information can be extracted from the T.F.AF, which is called the MTVC. The MTVC contains detail information of SHM that enable observers to read the structural modal behavior directly without traditional mechanical calculation. It is a new and crucial way for us to explore useful SHM information from the real building records. Examining these MTVCs can explore the vivid differences of structure healthy condition that might be ignored by other observers.
The specialities of the new method are the following three: 1. The results are totally obtained by the vibration data, it’s an adaptive and data-driven method. 2. The structural and mechanical concepts of the theoretical linear and stationary governing equations are included in the new method. 3. The traditional linear and stationary model and modal calculations are not included in the new method. We had demonstrated the excellent performances of the new method. The vivid differences of the MTVCs from different earthquake events showed the corresponding slight structural differences of a healthy building.
The T.F.AF is a new-created tool for the observation for a simple time-variant system. The energy density and instantaneous frequency distribution pattern on the time-frequency spectrum are the curical information for the observers. The vivid differences of the system can be easily found from T.F.AF. Now, the vibration-based results are having a drastic differnce with the theoretical solution. With new-created T.F.AF and the new HHT SHM method, useful dynamic information is explored and showed. These results might be helpful for the academic improvement of the basic mechanical model in the future. We showed a lot of observational results from real building and shaking-table experiment in the article, including SDOF and MDOF structures. Some of the behaviors are case unique and some are common phenomenon. We introduce a new observation tool; wish this helps to improve the performance of theoretical model in the future by researchers.
Keywords: Time-frequency analysis, Hilbert-Huang Transform, Strong-motion building response analysis, Vibration-based analysis, Structural Health Monitoring, Two-station method, Time frequency amplification function, Modal temporal variation curve.