# Orbit Decomposition Method for Rotordynamic Coefficients Identification of Annular Seals
## 2 Review of Transient Method for Rotordynamic Coefficient Identification  
Before going into the details of the orbit decomposition method for rotordynamic coefficient identification, it is necessary to introduce the traditional transient method. Figure 2 shows the elliptical orbit whirl model. The rotor performs an elliptical orbit whirl at a speed of \(\Omega\) around the stator center \(o_{s}\), and rotates about its own axis at a speed of \(\omega\). This whirl orbit is expressed as:  
\[\left\{ \begin{array}{l}X=a\cos(\Omega t)\cos(\alpha)-b\sin(\Omega t)\sin( \alpha)\\ Y=a\cos(\Omega t)\sin(\alpha)+b\sin(\Omega t)\cos(\alpha)\end{array} \right. \tag{1}\]where \(X\) and \(Y\) are the \(x\) and \(y\) direction rotor displacements, respectively, \(\Omega\) is the rotor whirl velocity, \(t\) is the time, \(\alpha\) is the ellipse attitude angle, and \(a\) and \(b\) are the major semiaxis and minor semiaxis, respectively. As the rotor location and fluid domain change with time, transient analysis and the moving mesh technique are required. A whirl period is divided into multiple time steps. The three-dimensional unsteady flow field is calculated step by step. In general, the transient computation should be run for several whirl cycles until the accurate transient solution is reached. The transient analysis is very time-consuming. After the time-dependent fluid-induced forces are obtained, the rotordynamic coefficients can be determined. For small orbital motion of the rotor about the stator center, the fluid-induced forces are quantified by specifying a set of linearized rotordynamic coefficients, as shown in the following equation [23]:  
\[-\left[ \begin{array}{c}F_{x}\\ F_{y}\end{array} \right]=\left[ \begin{array}{cc}K(\Omega)&k(\Omega)\\ -k(\Omega)&K(\Omega)\end{array} \right]\left[ \begin{array}{cc}X\\ Y\end{array} \right]+\left[ \begin{array}{cc}C(\Omega)&c(\Omega)\\ -c(\Omega)&C(\Omega)\end{array} \right]\left[ \begin{array}{c}\frac{dX}{dt}\\ \frac{dt}{dt}\end{array} \right] \tag{2}\]  
where \((F_{x},F_{y})\) are the \(x\) and \(y\) direction components of the fluid-induced force acting on the rotor, and \((K,k)\) and \((C,c)\) are the stiffness and damping coefficients, respectively. All the coefficients are functions of rotor whirl velocity \(\Omega\). Substituting Equation (1) into Equation (2), the force-displacement relation of the elliptical orbit model is expressed as:  
\[ \begin{array}{c}-\left[\begin{array}{c}F_{x}\\ F_{y}\end{array}\right]=\left[\begin{array}{cc}K(\Omega)&k(\Omega)\\ -k(\Omega)&K(\Omega)\end{array}\right]\left[\begin{array}{cc}a\cos(\Omega t )\cos(\alpha)-b\sin(\Omega t)\sin(\alpha)\\ a\cos(\Omega t)\sin(\alpha)+b\sin(\Omega t)\cos(\alpha)\end{array}\right]\\ +\left[\begin{array}{cc}C(\Omega)&c(\Omega)\\ -c(\Omega)&C(\Omega)\end{array}\right]\left[\begin{array}{cc}-a\Omega\sin( \Omega t)\cos(\alpha)-b\Omega\cos(\Omega t)\sin(\alpha)\\ -a\Omega\sin(\Omega t)\sin(\alpha)+b\Omega\cos(\Omega t)\cos(\alpha)\end{array}\right] \end{array}  \tag{3}\]  
For each whirl frequency, the cases of \(\Omega t\) = 0 rad and \(\Omega t\) = \(\pi/2\) rad are selected to identify the frequency-dependent rotordynamic coefficients. Substituting \(\Omega t\) = 0 rad into Equation (3), the force-displacement relation is expressed as:  
\[-\left[ \begin{array}{c}F_{x}^{0}\\ F_{y}^{0}\end{array} \right]=\left[ \begin{array}{cc}K(\Omega)&k(\Omega)\\ -k(\Omega)&K(\Omega)\end{array} \right]\left[ \begin{array}{cc}a\cos(\alpha) \\ a\sin(\alpha)\end{array} \right]+\left[ \begin{array}{cc}C(\Omega)&c(\Omega) \\ -c(\Omega)&C(\Omega)\end{array} \right]\left[ \begin{array}{cc}-b\Omega\sin( \alpha)\\ b\Omega\cos(\alpha)\end{array} \right] \tag{4}\]  
where \((F_{x}^{0},\,F_{y}^{0})\) are the \(x\) and \(y\) direction fluid-induced force components of the elliptical orbit model in the case of \(\Omega t\) = 0 rad. Figure 2: Elliptical orbit whirl model. The blue circle and red circle are auxiliary circles that help to locate the position of the whirl rotor.