本文應用傅立葉正弦頻譜和傅立葉正弦時頻圖產生器，對一架遙控電動直升機和無人遙控飛行載具的聲音數據，以及滾珠軸承的振動數據做分析討論。時頻產生器包括：使用疊代式高斯平滑法移除非週期和極低頻部份，再用單調式三次方分段內插法，幫忙定出所剩的週期性部份數據兩端的零點，經內插重新分佈點及奇函數轉換，以快速傅立葉級數轉換法，求取傅立葉正弦頻譜後。隨後對頻譜取有限的帶狀頻帶後，對頻譜做快速傅立葉級數逆轉換，求得傅立葉正弦時頻圖之實數部份，再應用希爾伯特轉換式求振幅並畫出時頻圖。而從時頻圖中可以詳細而正確地看到一堆模組(modes)和其諧波(harmonics)，其頻率跟振幅隨著時間變動，因此可以從中檢測出是否有異常情形。本文將對遙控電動直升機分別做主旋翼、尾旋翼及整架直升機之手持狀態做分析。而無人遙控飛行載具部份將單就對其引擎測試，然後實機地面測試最後實地飛行。最後滾珠軸承將對正常狀態、內環損傷、滾珠損傷和外環損傷做時頻分析比較。這些測試顯示本文所用的頻譜和時頻圖，具有可以作為監測完整系統的一種工具之潛力。

The Fourier sine spectrum and Fourier sine spectrogram generators are employed to examine acoustic data of a remote control unmanned electrical helicopter, an Unmanned Aerial Vehicle (UAV) and vibration data of ball bearing. The transformation contains the following steps. For a given data string whose non-sinusoidal part have been removed, zero points around the two ends are identified by searching and interpolation. After dropping segments beyond two zero ends, the corresponding Fourier sine spectrum is obtained. The time-frequency transform imposes finite bandwidth Gaussian window upon the Fourier sine spectrum centered at a given frequency. The inverse Fourier transform of the band-pass limited spectrum gives the real part of the resulting spectrogram. Form the resulting spectrums much information concerning the dominate modes and corresponding harmonics can be captured. Moreover, the associated amplitude and frequency variations are partially reflected by the spectrum scattering around a mode. The detailed variations can be obtained by examining the spectrogram. This study separately examined the acoustic data of main rotor wing, tail rotor wing and holding by hand of the unmanned electrical helicopter. Many detailed information can be precisely extracted. For the UAV, the engine test information, the ground test information and take-off operation are examined. The result shows that the engine test facility is too rough. For the bearing test, defects of the inner race, rolling element and outer race are compared with the normal state. The resulting spectrum and spectrogram show much new information. These tests show that the spectrum and spectrogram have the potential to monitor the operating state of a complete system.

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Case Western Reserve University Bearing Data Center Website