# Identification of Impact Frequency for Down-the-Hole Drills Using Motor Current Signature Analysis
## 3 Results
### Envelope Approach  
In signal processing, the envelope function of an oscillating signal is a curve outlining the extremes of the signal in a smooth way. Envelope functions provide a convenient way to represent instantaneous changes in the amplitude of an oscillating signal, and it is then suitable for analyzing the changes in the amplitude of the motor current observed during the SP-stage. The envelopes of the motor current signal are computed using spline interpolation over the local maxima (i.e., current peaks), separated by \(n\) samples. The  
Figure 10: Impact frequency identification from waterfall plots during stage-z: (**a**) I\({}_{m,fund}\) signal; (**b**) zoom of I\({}_{m,fund}\) signal; (**c**) frequency spectrum of I\({}_{m,fund}\). Figure 9: Waterfall plot for motor current during stage-z. maximum number of spline interpolation samples (\(n_{max}\)) between successive current peaks is obtained by  
\[n_{max}=\frac{f_{s}}{f_{f}} \tag{1}\]  
where \(f_{s}\) is the sample frequency and \(f_{f}\) is the fundamental frequency of the motor current. For our case study, \(f_{s}\) is 50 kHz and \(f_{f}\) is 48.2 Hz (see Figure 8), resulting in a \(n_{max}=1037\) that represents an upper-bound for \(n\). To illustrate the effects of the choice of the number of samples \(n\) used for spline interpolation in the shape of the resulting envelope, Figure 11 shows the motor current signal during a window of the SP-stage, including the upper (red line) and lower (yellow line) envelopes considering three values of \(n\). The depicted examples consider \(n=530\) (lower than \(n_{max}\)), \(n=970\) (close to \(n_{max}\)), and \(n=1200\) (greater than \(n_{max}\)). The plot shows that setting a low value of \(n\) produces a distorted envelope that does not capture the amplitude peaks of the current signal. On the other hand, when the number of samples is greater than \(n_{max}\), the envelope signal cannot consistently capture the current peak modulation, generating artificially higher amplitude peaks. When considering several samples close to \(n_{max}\), the resulting envelopes accurately outline the peaks of the motor current. To illustrate the effect of the chosen \(n\) over the spectra of the resulting envelopes, Figure 12 shows the frequency spectra for the three values of \(n\) presented above. Figure 12 shows that impact frequency is identified with a certain accuracy grade regardless of the number of samples used for the spline interpolation. However, the amplitude of both the motor current at the impact frequency and the artificial frequency content are significantly affected by \(n\). For \(n=530\), the noticeable distortions in the envelope shape and the overestimation of the peaks of the current signal amplify the current amplitude at the impact frequency and generate artificial frequency components comparable in amplitude to the impact frequency. As expected, for \(n=970\), the impact frequency is clearly distinguishable, and the amplitude coincides with the magnitude of the current. In the case of \(n=1200\), the  
Figure 11: Upper (red line) and lower (yellow line) envelopes of motor current signal (blue line) during SP-stage with different number of samples for spline interpolation: (**a**) 530 samples; (**b**) 970 samples; (**c**) 1200 samples. impact frequency is still clearly distinguishable in the spectra, but the overestimation of the peaks of the current signals represented in the envelope generates a higher amplitude for the impact frequency and artificial frequency content. The observed results suggest that the method for frequency identification using the envelope approach is not very sensitive to the choice of the parameter \(n\). Indeed, repeating the procedure described above for different values of \(n\) and performing a sensitivity analysis over the identified frequency, we determined that, for our case study, setting the number of samples in the range \((0.53\times n_{max})\leq n\leq(0.93\times n_{max})\) permitted us to identify the impact frequency.