# Orbit Decomposition Method for Rotordynamic Coefficients Identification of Annular Seals
## 3 Rotordynamic Coefficient Identification Based on Orbit Decomposition Method
### Fluid-Induced Force Superposition  
Substituting Equation (8) into Equation (2), the fluid-induced forces \((F_{x},F_{y})\) of the elliptical orbit model are obtained as:  
\[ \begin{array}{l}-\left[\begin{array}{c}F_{x}\\ F_{y}\\ \end{array}\right]=\left[\begin{array}{cc}K(\Omega)&k(\Omega)\\ -k(\Omega)&K(\Omega)\\ \end{array}\right]\left[\begin{array}{cc}X_{f}+X_{b}\\ Y_{f}+Y_{b}\\ \end{array}\right]+\left[\begin{array}{cc}C(\Omega)&c(\Omega)\\ -c(\Omega)&C(\Omega)\\ \end{array}\right]\left[\begin{array}{c}\frac{d(X_{f}+X_{b})}{d(Y_{f}^{d}+Y_ {b})}\\ \frac{d(Y_{f}^{d}+Y_{b})}{dt}\\ \end{array}\right]\\ \end{array} \right.\\ =\left[ \begin{array}{cc}K(\Omega)&k(\Omega)\\ -k(\Omega)&K(\Omega)\\ \end{array} \right]\left[ \begin{array}{c}X_{f}\\ Y_{f}\\ \end{array} \right]+\left[ \begin{array}{cc}C(\Omega)&c(\Omega)\\ -c(\Omega)&C(\Omega)\\ \end{array} \right]\left[ \begin{array}{c}\frac{d(X_{f})}{d(Y_{f}^{d})}\\ \frac{d(Y_{f}^{d}+Y_{b})}{dt}\\ \end{array} \right]+\\ \left[ \begin{array}{cc}K(\Omega)&k(\Omega)\\ -k(\Omega)&K(\Omega)\\ \end{array} \right]\left[ \begin{array}{c}X_{b}\\ Y_{b}\\ \end{array} \right]+\left[ \begin{array}{cc}C(\Omega)&c(\Omega)\\ -c(\Omega)&C(\Omega)\\ \end{array} \right]\left[ \begin{array}{c}\frac{d(X_{b})}{d(Y_{b})}\\ \frac{d(Y_{b})}{dt}\\ \end{array} \right]\end{array}+\\ \left[ \begin{array}{c}\frac{d(X_{b})}{d(Y_{b})}\\ \frac{d(Y_{b})}{dt}\\ \end{array} \right]\end{array} \tag{9}\]  
For the forward circular orbit whirl model, the fluid-induced forces \((F_{x\!f},F_{y\!f})\) are expressed as:  
\[-\left[ \begin{array}{c}F_{x\!f}\\ F_{y\!f}\\ \end{array} \right]=\left[ \begin{array}{cc}K(\Omega)&k(\Omega)\\ -k(\Omega)&K(\Omega)\\ \end{array} \right]\left[ \begin{array}{c}X_{f}\\ Y_{f}\\ \end{array} \right]+\left[ \begin{array}{cc}C(\Omega)&c(\Omega)\\ -c(\Omega)&C(\Omega)\\ \end{array} \right]\left[ \begin{array}{c}\frac{dX_{f}}{dY_{f}}\\ \frac{dY_{f}}{dt}\\ \end{array} \right] \tag{10}\]  
Figure 4: Elliptical Orbit Decomposition. For the backward circular orbit whirl model, the fluid-induced forces (\(F_{xb}\), \(F_{yb}\)) are expressed as:  
\[-\left[ \begin{array}{c}F_{xb}\\ F_{yb}\end{array} \right]=\left[ \begin{array}{cc}K(\Omega)&k(\Omega)\\ -k(\Omega)&K(\Omega)\end{array} \right]\left[ \begin{array}{cc}X_{b}\\ Y_{b}\end{array} \right]+\left[ \begin{array}{cc}C(\Omega)&c(\Omega)\\ -c(\Omega)&C(\Omega)\end{array} \right]\left[ \begin{array}{c}\frac{dX_{b}}{ dt}\\ \frac{dX_{b}}{dt}\end{array} \right] \tag{11}\]  
Hence, the fluid-induced forces of the elliptical orbit model can be obtained by the superposition of fluid-induced forces due to the forward and backward circular orbit whirl models, as shown in Figure 5. The fluid-induced forces of the elliptical orbit model can be obtained by:  
\[\left[ \begin{array}{c}F_{x}\\ F_{y}\end{array} \right]=\left[ \begin{array}{c}F_{xf}\\ F_{yf}\end{array} \right]+\left[ \begin{array}{c}F_{xb}\\ F_{yb}\end{array} \right] \tag{12}\]  
In Figure 5 and Equation (12), the fluid-induced forces \(F_{x}\) and \(F_{y}\) are positive when they are in the positive directions of the \(x\)-axis and \(y\)-axis, respectively. Conversely the fluid-induced forces \(F_{x}\) and \(F_{y}\) are negative when they are in the negative directions of the \(x\)-axis and \(y\)-axis, respectively. For the circular orbit model, the tangential force \(F_{t}\) and radial force \(F_{r}\) acting on the rotor are constant at any circumferential position. According to the coordinate transformation shown in Figure 6, the time-dependent fluid-induced forces of the two decomposed circular orbit models can be expressed as:  
\[\left[ \begin{array}{c}F_{xf}\\ F_{yf}\end{array} \right]=\left[ \begin{array}{c}F_{rf}\cos(\Omega t+\alpha)- F_{tf}\sin(\Omega t+\alpha)\\ F_{rf}\sin(\Omega t+\alpha)+F_{tf}\cos(\Omega t+\alpha)\end{array} \right] \tag{13}\]  
and  
\[\left[ \begin{array}{c}F_{xb}\\ F_{yb}\end{array} \right]=\left[ \begin{array}{c}F_{rb}\cos(-\Omega t+\alpha)- F_{tb}\sin(-\Omega t+\alpha)\\ F_{rb}\sin(-\Omega t+\alpha)+F_{tb}\cos(-\Omega t+\alpha)\end{array}