# Identification of Close Modes on Frequency in Rotating Systems
## 2 Close modes in Frequency  
### Mathematical Model  
In the present study, the system is considered as shown in the following Figure 1:Where \(M\) represents the total mass of the system, \(m_{j}\) the imbalance mass, \(c\) the viscous damping coefficient, and \(k\) the stiffness. Consider that the imbalance force is given by the following relationship:  
\[F_{des}\ \alpha\ \omega^{2} \tag{1}\]  
Therefore, the force equation is \(f(t)\ =\ m_{j}\ R\ \omega^{2}\ e^{i\omega t}\), where \(R\) represents the radius of placement of the imbalance mass.  
Applying [PERSON]'s second law to the diagram shown in Figure 1 and considering the behavior of the system in a steady state returns the following equation:  
\[M\ \tilde{x}\ +\ c\ \tilde{x}\ +\ k\ x\ =\ f(t), \tag{2}\]  
Considering a solution of the form \(x\ =\ X\ e^{i\ \omega\ t\ -\ \phi}\), the system response is given by the following equations:  
\[X\ =\ \frac{m_{j}\ R\ \omega^{2}}{\sqrt{\left(k\ -\ M\omega^{2}\right)^{2}\ +\ \left(c \omega\right)^{2}}}, \tag{3}\]  
while the phase angle is given by  
\[\phi\ =\ tg^{-1}\bigg{(}\frac{c\omega}{k\ -\ M\omega^{2}}\bigg{)}, \tag{4}\]  
Using Equations (3) and (4) allows generating polar response diagrams from a decoupled vibration mode. However, the coupled dynamic response of a rotating system can be generated from these equations, as presented in [3].  
Likewise, the methodology for two-degree-of-freedom coupled rotating systems is presented in matrix form in [4; 16].  
For the present research work, modal analysis will be used to generate the dynamic response of a decoupled system rotor to graphically represent the definition of close modes and modes separated in frequency. To do this, the expressions presented in [17] are used, given as  
\[{}_{i}X^{\omega}_{real}\ =\ R_{j}\ m_{j}\ \sum_{r=1}^{t}\frac{{}_{r}\psi_{j}\,{}^{r} \psi_{i}\,\Omega_{r}^{2}\ \left(1\ -\ \Omega_{r}^{2}\right)}{MM_{r}\left(\left(1\ -\ \Omega_{r}^{2}\right)^{2}\ +\ \left(2\ \zeta_{r}\ \Omega_{r}\right)^{2}\right)}\, \tag{5}\]  
\[{}_{i}X^{\omega}_{imag}\ =\ R_{j}\ m_{j}\ \sum_{r=1}^{t}\frac{{}_{r}\psi_{j}\,{}^{r} \psi_{i}\,2\,\zeta_{r}\ \Omega_{r}^{3}}{MM_{r}\left(\left(1\ -\ \Omega_{r}^{2}\right)^{2}\ +\ \left(2\ \zeta_{r}\ \Omega_{r}\right)^{2}\right)}\, \tag{6}\]  
where \({}_{i}X^{\omega}_{real}\), \({}^{i}X^{\omega}_{imag}\) represent the real and imaginary components of vibration observed by a sensor at position \(i\) given an angular velocity \(\omega\), \({}^{r}\psi_{j}\) is the modal form factor of mode  
Figure 1: Simplified model of a 1-DOF system with a force due to imbalance.  
\(r\) observed by a sensor at position \(j\), \(MM_{r}\) is the modal mass of mode \(r\), \(\zeta_{r}\) is the viscous damping factor of mode \(r\), and \(\Omega_{r}\) is the frequency ratio of mode \(r\).  
Equations (5) and (6) allow generating decoupled vibration modes horizontally and vertically to generate the \"mode pairs\" shown in [15]. Likewise, the methodology proposed in [17] is used to generate the coupling for the generation of polar diagrams from more than two coupled modes.  
Considering two vibration modes, the numerical parameters presented in the table below will be used.  
The polar diagram of the corresponding mode pair observed by the sensor at position 1 in the 0 and 90\({}^{\circ}\) directions generated with Equations (5) and (6) returns the Figure 2.  
Figure 2a,b presented the decoupled modes in \(x\) and \(y\) direction using the parameter shown in Table 1.