# Measurement and Analysis of Downhole Drill String Vibration Signal
## 3 Result Analysis
### Frequency Analysis of Vibration Signal  
By analyzing the time domain signal, the vibration pattern and intensity of the drill string can be intuitively judged, but the frequency information of the drill string vibration cannot be judged. Through the frequency spectrum analysis and time-frequency analysis of the vibration information, the vibration frequency information of the drill string can be understood in more detail.  
The frequency analysis of the drill string vibration signal mainly uses two methods, fast Fourier transform (FFT) and short-time Fourier transform (STFT) [36; 37]. Fast Fourier transform can analyze each frequency component in drill string vibration, and short-time Fourier transform can clarify the change law of each frequency with time.  
Fast Fourier Transform (FFT) is a signal-processing method proposed on the basis of Discrete Fourier Transform (DFT). For a signal sequence of length \(L\), the form expressed by FFT is:  
\[X_{L}(n)=\left\{\begin{array}{cc}X_{\
icefrac{{L}}{{2}}}^{e}(n)+W_{L}^{n}X_{ \
icefrac{{L}}{{2}}}^{o}(n)&n=0,1,\dots\
icefrac{{L}}{{2}}-1\\ X_{\
icefrac{{L}}{{2}}}^{e}(n-L)-W_{L}^{n-\
icefrac{{L}}{{2}}}X_{\
icefrac{{ L}}{{2}}}^{o}(n-\
icefrac{{L}}{{2}})&n=\
icefrac{{L}}{{2}},\dots,L-1\end{array}\right. \tag{7}\]  
where \(X_{L}\) is the determined result with \(N\) data points after DFT, \(X_{\
icefrac{{L}}{{2}}}^{e}\) is the DFT result of the even-numbered branch with \(\
icefrac{{L}}{{2}}\) points, and \(X_{\
icefrac{{L}}{{2}}}^{o}\) is the DFT result of the odd-numbered branch with \(\
icefrac{{L}}{{2}}\) points. Through Fast Fourier Transform (FFT), you can intuitively recognize the frequency distribution of the drill string vibration signal and the energy of each main frequency.  
The short-time Fourier transform (DFT) is a time-frequency analysis method. This method uses a sliding time window to stage a signal so that it can be approximated as a short-term stable signal in this time window. The time-stationary signal undergoes Fourier transform to obtain the time-frequency distribution characteristics of this signal. The short-time Fourier transform is expressed as:  
\[F_{STFT}(t,f)=\int_{-\infty}^{+\infty}x(t)[g(t-\tau)e^{-j2\pi f\,t}]dt \tag{8}\]  
where \(x(t)\) is the time domain signal, \(g(t-\tau)\) is the window function, \(f\) is the frequency, and \(F_{STFT}(t,f)\) is the transformed time-frequency signal. In the short-time Fourier transform process, the longer the time window, the better the frequency resolution and the worse the time resolution. Once the window function is determined, it means that the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Vibration Pattern** & **Vibration Characteristics** & **Criterion** \\ \hline \multirow{8}{*}{Stick-slip} & & There are periodic fluctuations in the \\  & & radial acceleration in the mean, root \\  & Periodic ’sticky-slippage’ of & mean square, and peak curves, and the \\  & the drill string & fluctuation period is roughly the same \\  & & as the wave packet \\  & & ’appear-disappear-appear’ period in the \\  & & time domain signal. \\  & & The mean value of radial acceleration is \\  & & quite different from the mean value of \\  & Coupling of lateral vibration & tangential acceleration, and the mean \\  & and torsional vibration & value of tangential acceleration is not \\  & & zero. The coincidence of the root mean \\  & & square and peak curve is better. \\ \hline \hline \end{tabular}
\end{table}
Table 5: Drill string vibration modes and criteria.  
resolution of the STFT is also determined and unchanged; so, the selection of the window function must take into account the resolution of time and frequency.