# Measurement and Analysis of Downhole Drill String Vibration Signal
## 2 Principle and Method of Drill String Vibration Measurement
### Vibration Signal Processing  
The drill string rotates and revolves downhole. Because the sensor is installed eccentrically on the measuring equipment, the vibration acceleration at the center of the drill string has a certain relationship with the vibration acceleration collected by the sensor. Assuming that the installation eccentricity of the sensor on the measuring equipment is \(R\), the relationship between the measured value of the orthogonal three-axis acceleration sensor in three directions and the center acceleration of the drill string is:  
\[a_{SX}=a_{CX}+R\cdot\dot{\omega} \tag{1}\]  
\[a_{SY}=a_{CY}+R\cdot\omega^{2} \tag{2}\]  
\[a_{CZ}=a_{SZ} \tag{3}\]  
where \(a_{CX}\) and \(a_{CY}\) are the lateral vibration acceleration of the drill string center, \(a_{CZ}\) is the axial vibration acceleration of the drill string center, \(a_{SX}\) is the vibration acceleration of the sensor center along the tangential direction of the drill string, \(a_{SY}\) is the acceleration of the sensor center along the drill string radial direction, and \(a_{SZ}\) is the acceleration of the sensor center along the drill string axial direction. The unit of acceleration is the acceleration of gravity \(g\), \(R\) is the distance from the sensor to the center of the drill string, and the unit is \(m\). \(\omega\) is the rotation angular velocity of the drill string, a function of time, and the unit is \(rad/s\). There are four basic patterns of drill string downhole vibration: axial vibration, lateral vibration, torsional vibration, and whirl. During drilling, these drill string vibration patterns may alternately occur or complex coupled vibrations may occur. The vibration acceleration signal collected by the vibration measurement device cannot quantify the specific vibration pattern. However, by analyzing the average value, root mean square value, and peak value of the measured value from the sensor in the time domain, the main vibration mode of the drill string in this time interval can be qualitatively judged. When the drill string vibrates axially, the drill bit will jump up and down, which will seriously damage the PDC drill bit. It can be judged directly by the peak value and fluctuation amplitude of the axial acceleration signal. Assume that the average values of the tangential acceleration and the radial acceleration measured by the sensor in a certain time window are \(\overline{a}_{SX}\) and \(\overline{a}_{SY}\). The rotation speed of the drill string is constant when general lateral vibration occurs. The integrals of \(a_{SX}\) and \(a_{SY}\) in one movement cycle are equal, so the value of \(\overline{a}_{SY}-\overline{a}_{SX}\) is very small when the drill string undergoes general lateral vibration. Torsional vibration is a vibration mode produced by the interaction between the drill bit and the formation or the drill string and the borehole wall. When a single torsional vibration occurs, the speed changes more drastically, making \(R\cdot\omega^{2}\) much larger than \(R\cdot\dot{\omega}\), so the value of \(\overline{a}_{SY}-\overline{a}_{SX}\) is larger when torsional vibration occurs. Stick-slip vibration is a self-excited torsional vibration. When the downhole drill string undergoes Stick-slip vibration, the collected triaxial acceleration signals will show regular intermittent wave packets in the time domain. At this time, the rotation speed of the downhole drill string will appear in two states: stagnation and slippage. Periodic peaks and valleys also appear on the mean value, root mean square value, and peak curve. The whirling of the drill string has been proven to be one of the main reasons for the premature failure of the PDC drill bit [13]. When the drill string whirl occurs, the acceleration frequency of the drill string lateral vibration will be proportional to the drill string rotation speed, and the amplitude of the radial acceleration and the tangential acceleration are roughly the same [35]. The difference between ASX and ASY is relatively large, and ASX will have a large peak.