# Identification of Close Modes on Frequency in Rotating Systems
## 2 Close modes in Frequency
### Identification of Close Modes in Frequency  
For the process of identifying close modes in frequency in rotating systems, the extreme case is proposed, in which the two natural frequencies of the horizontal and vertical mode are equal, as presented in Figure 5. Equation (7) will be applied to generate a 10\({}^{\circ}\) scan in different angular positions of the imaginary sensor \"\(q\)\" in the range of 0 to 180\({}^{\circ}\), taking the horizontal and vertical vibration modes generated with Equations (5) and (6), respectively. This will generate 19 numerical signals for this particular case, which are denominated synthetic signals in this article. In order to generalize the methodology for identifying close modes in frequency in rotating systems, \"m\" synthetic signals will be generated by applying the imaginary sensor equation. Next, the modal parameters (natural frequency, phase angle, damping ratio, and vibration amplitude in displacement) will be extracted from each of the \"m\" synthetic signals using the adjustment method presented in [3], by adjusting circles and programming the AMODAL software. An example of the modal parameter extraction for the synthetic signal given at the angular position \(\alpha\ =\ 20^{\circ}\) is presented in the Figure 6:  
1 Modes: - Ajustes conamoriguamiento viscososo, en Integral: No. omega Amort. Magnitud de Fase Di metro [rad/s] [pu] la constante (\(\phi\)) 1M 150.800 0.049997 22.5545 -159.961 1.49575  
Subsequently, each of the modal parameters is plotted in relation to the variation of the imaginary sensor obtaining the data for each synthetic signal as shown in Figure 6, generating the graphs \(\omega_{n}\ vs\ \alpha\), \(\phi_{\Omega\ =\ 1}\ vs\ \alpha\), \(\zeta\ vs\ \alpha\), \(X_{\Omega\ =\ 1}\ vs\ \alpha\) as shown in the Figure 7.  
The Figure 7a-d, allows identification of the close modes problem, where at least two vibration modes are present, is the \(\phi_{\Omega\ =\ 1}\ vs\ \alpha\), showing an apparent improvement of the phase angle with respect to each change of the sensor's angular position. The previous case is an ideal case; however, a second case study considers a 1.38% separation between the frequency of the horizontal and vertical mode. For this, most parameters shown in Table 1 are considered, now taking into account that \(\omega_{n\ x}\ =\ 150.79\ \mathrm{rad/s}\)  
Figure 6: Adjusted modal parameters when an imaginary sensor has an angular position of 20\({}^{\circ}\). Figure 7: Characteristic patterns of close modes in frequency considering \(\Delta\omega\ =\ 0\%\) for (**a**) natural frequency, (**b**) damping ratio, (**c**) phase angle and (**d**) vibration amplitude at resonance. and \(\omega_{n~{}y}~{}=~{}148.70\) rad/s. Applying the same steps proposed for the previous case, the following patterns are presented in Figure 8.  
For this particular case study, Figure 8c \(\phi_{\Omega\,=\,1}\) vs. \(\alpha\) again allows identifying the problem of close modes in rotating systems when there is a \(1.38\%\) frequency variation. Additionally, a periodic variation in the frequency is presented in Figure 8a \(\omega_{n}\) vs. \(\alpha\), where the maximum and minimum of this variation is close to the natural frequency of each mode. Finally, the proposed methodology was applied in a field turbogenerator using displacement vibration sensors with a sensitivity of \(100\frac{\mathrm{m}}{\mathrm{g}}\), where from the analysis carried out in each of the eight bearings, bearing number 2 presented patterns of interest for the present investigation in one of the identified modes. Therefore, the polar diagram measured by the rotor instrumentation placed at 45\({}^{\circ}\) and 135\({}^{\circ}\) is used, as shown in the following Figure 9.  
In the previous figure, the red spot at the point is observed at the beginning of the vibration measurement and the equipment's considerable run-out.