## 3 Results
### Proposed PSEMDI Method  
The proposed method was validated by comparing the velocity obtained using the PSEMDI with the signal measured directly by a laser vibrometer, and both devices were synchronized. The laser vibrometer focused on the disk surface close to the friction point. Due to the measurement points being different in position and orientation, the vibrometer signal had to be corrected in order to be compared. The input signal of the unbalanced disk with friction at 700 rpm is represented by Figure 10a. The velocity curves were compared as shown in Figure 10b, and then the coherence between both signals was calculated and it is presented in Figure 10c. The primary frequencies were located at the peaks A (11.72 Hz, 0.22), B (152.3 Hz, 0.14), and C (199.2 Hz, 0.13), where point A is the angular speed at  
Figure 8: Transformation of acceleration into velocity by detrending with the SDEMDI method for Case 1 at 700 rpm: (**a**) Acceleration signal; (**b**) velocity resulting from integration with its trend; (**c**) corrected velocity and displacement.  
Figure 7: (**a**) Comparison of signal for three different disk conditions at 900 rpm: free, unbalanced, and unbalanced with friction; (**b**) friction behavior for different speeds.  
700 rpm and the others represent friction frequencies. The velocity signals compared in Figure 10b have a significantly similar pattern despite the low values of the coherence.  
The PSEMDI procedure is described with two intrinsic mode functions for Case 1 at 700 rpm, according to the block diagram in Figure 6. IMF3 and IMF8 were chosen from results in Figure 11. For each \(IMF\), the FFT was computed and its fundamental frequency and amplitude were calculated. The fundamental frequency is used to phase shift the \(IMF\), and the amplitude was used to correct signal amplitude. Then, the velocity was estimated  
Figure 10: Comparison of PSEMDI velocity with laser vibrometer for the unbalanced disk and friction at 700 rpm: (**a**) Acceleration signal; (**b**) comparison of PSEMDI velocity and vibrometer; (**c**) coherence results.  
Figure 9: \(IMF\)s obtained from acceleration by EMD.  
by shifting the \(IMFs\) by 1/4 of the fundamental frequency. These steps are shown in Figure 11 for two of the \(IMFs\).  
Finally, the \(IMFs\) were computed and added to obtain the final velocity and displacement. The penultimate \(IMF\) provides the maximum amount of data for the calculation. The results of the calculation of the velocity and displacement with the PSEMDI method are shown in Figure 12. The displacement has the smallest trend, and all vibrations are symmetrical and stable. The results are very different from those obtained with the Simpson's rule integration. The most interesting difference in the results is that there is no significant trend and, as a consequence, the spectra are more symmetric and stable in contrast with the SDEMDI.  
Figure 11: The phase shift property applied on two \(IMFs\) (Case 1, 700 rpm): (**a**) \(IMF3\); (**b**) \(IMF8\); their fundamental FFT frequency and its amplitudes are (**c**,**d**), respectively. The phase shift are 1/4 of the fundamental frequency for velocity and displacement, respectively.