# Identification of Close Modes on Frequency in Rotating Systems
## 2 Close modes in Frequency
### Signals Coupling Using Imaginary Sensor  
The method presented in [3; 18] is used to perform the coupling of the two vibration modes shown in Figure 2 through the following equation:  
\[q\ =\ \frac{x^{\prime}sin(\beta-\alpha)\ +\ x^{\prime\prime}sin(\alpha\ -\ \theta)}{sin(\beta\ -\ \theta)}, \tag{7}\]  
where  
\[ \begin{array}{lcl}x^{\prime}&=&X_{c}^{\prime}\ \cos(\omega t)\ +\ X_{s}^{ \prime}\ sin(\omega t),\\ x^{\prime\prime}&=&X_{c}^{\prime\prime}\ \cos(\omega t)\ +\ X_{s}^{\prime \prime}\ sin(\omega t)\end{array}  \tag{8}\] \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Quantity** \\ \hline Modal mass of the first mode (\(M\ M_{r}\)) & 55,337.11 (kg) \\ Horizontal mode natural circular frequency (\(\omega_{n\ x}\)) & 159.79 (rad/s) \\ Horizontal mode damping ratio (\(\zeta_{r\ x}\)) & 0.05 \\ Vertical mode natural circular frequency (\(\omega_{n\ y}\)) & 129.85 (rad/s) \\ Vertical mode damping ratio (\(\zeta_{r\ y}\)) & 0.05 \\ Imbalance mass (\(m_{j}\)) & 2 to 0\({}^{\circ}\) (kg) \\ Imbalance mass placement radius (\(R\)) & 0.45 (m) \\ Angular velocity (\(\omega\)) & 0–188.49 (rad/s) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters for modeling the first vibration mode pair. Figure 2: Decoupled vibration modes using 1-DOF Systems by (**a**) horizontal and (**b**) vertical directions. and where \(\alpha\) represents the angular position of an imaginary sensor exhibiting the coupling of horizontal and vertical modes; \(q\) is the amplitude of vibration of the imaginary sensor given a desired position \(\alpha\); \(\theta\) is the angle between the horizontal axis and the direction \(x\prime\), which for the particular case of \(\theta~{}=~{}0^{\circ}\); and \(\beta\) is the angle between the horizontal axis and the direction \(x^{\prime\prime}\), which for the particular case of \(\beta~{}=~{}90^{\circ}\). This can be seen in a simplified image of a bearing, as presented in Figure 3.  
In Figure 3, \(S_{x}\) is the horizontally placed sensor that detects the rotor's dynamic response. In the initial case study, \(S_{x}\) detected the rotor response generated using Equations (5) and (6), as presented in Figure 2, with the sensor at \(0^{\circ}\), while \(S_{y}\) is the vertically placed sensor, whose behavior is shown in Figure 2 with the sensor at \(90^{\circ}\). Finally, \(q\) detects the dynamic response that a sensor would observe physically; however, Equation (7) is a numerical treatment from the coordinate transformation for obtaining the same result. It is common practice for the vibration sensors installed in the rotating systems in the field to be at \(45^{\circ}\) and \(135^{\circ}\), thus obtaining the coupling of two separate modes, as presented in Figure 4.  
The vibration mode pair separated in frequency is clearly visible in Figure 4a. Likewise, for the sensor located at \(135^{\circ}\), the presence of two vibration modes is clearly identified, as shown in Figure 4b. In field balancing processes, it is common to find separate modes, as presented in [16; 19; 20; 21; 22], by using Bode diagrams. For rotating systems, a close mode in frequency can be defined when there is a combination of at least two modes in the polar response diagrams whose natural frequency relationships--ratio to damping, vibration amplitude ratio, and phase angle--are such that only one vibration mode is apparently generated in the overall system response. The  
Figure 4: Coupling of 2 DOF vibration modes for a sensor position at (**a**) \(45^{\circ}\) and (**b**) \(135^{\circ}\). Figure 3: Simplified diagram of a bearing with mounted sensors. problem of close modes is commonly present in structures, leading to intense research to identify these modes, with the aim of monitoring the structure state or identifying problems, as presented in [6; 7; 8]. However, in these particular systems, the damping ratios are relatively low, ranging from 0.002 to 0.004, as shown in [7].