# Measurement and Analysis of Downhole Drill String Vibration Signal
## 3 Result Analysis
### Frequency Analysis of Vibration Signal
#### 3.2.1 Stick-Slip Vibration  
As shown in Figure 10, the frequency spectrum can be obtained by analyzing the measured stick-slip vibration signal using the fast Fourier (FFT) method. It can be seen from Figure 10 that the variety of signals with different frequencies of the radial acceleration is the largest, followed by the tangential acceleration, and the axial acceleration spectrum is the most single. In the frequency spectrum of radial acceleration and tangential acceleration, peaks appeared at 145.9 Hz, 155.6 Hz, and 216.5 Hz, indicating that the drill string had coupled vibration at these frequencies. In the frequency spectrum of 0-1 Hz in Figure 10, the main frequency components of the three-axis vibration acceleration are all at 0.1221 Hz, which is roughly equal to the vibration period of 10 s of stick-slip vibration. The 0.3967 Hz that appeared in the radial acceleration is about the triple frequency of 0.1221 Hz. In Figure 10, the main frequency amplitude of radial acceleration is the largest, and the main frequency amplitudes of tangential acceleration and axial acceleration are similar and both are smaller than radial acceleration. It shows that the centripetal acceleration caused by the change of the rotational speed in the stick-slip vibration had a significant influence on the radial acceleration. It further shows that in stick-slip vibration, the energy of torsional vibration is greater than lateral vibration and axial vibration.  
The short-time Fourier method (STFT) is used to analyze the signal of stick-slip vibration, as shown in Figures 11-14, and the relationship between time and frequency in the analysis signal is analyzed.  
Figure 10: Three-axis acceleration frequency spectrum of stick-slip vibration.  
It can be seen in Figures 11 and 12 that the two frequencies of 145.9 Hz and 155.6 Hz in the radial acceleration always exist in this segment of the signal. The light and dark lines in Figure 11 indicate the appearance and disappearance of each frequency, which can intuitively reflect the periodic appearance of other frequencies below 200 Hz, and the period of each appearance and disappearance is about 10 s. This also shows that the period of stick-slip vibration is 10 s. The time-frequency diagram of the tangential acceleration in Figure 13 shows that the two frequencies of 145.9 Hz and 155.6 Hz also always exist in this segment of the signal, and there are also high-frequency signals higher than 200 Hz. The regularity of other frequencies is weaker than that of radial acceleration, which also shows that radial acceleration is more sensitive to the occurrence of stick-slip vibration. In Figure 14, the two main frequencies of 145.9 Hz and 155.6 Hz also appear in axial vibration, which is a consequence of the fact that the frequency of stick-slip vibration is 10 s. This is because the frequency of stick-slip vibration is 10 s.  
and there are very few frequencies in other frequency bands of axial vibration. This shows that the axial vibration is weak during the stick-slip vibration process.