# Comparing the Performance of Robust Controllers for Vibration Suppression in Long Rotor Systems
## 2 Open-Loop Transfer Function Matrix Estimation  
A rotor can be idealized as \(n+1\) of the distributed parameter shafts. Those shafts are connected via \(n\) lumped parameter disks. The overall assembly is mounted on bearings and is shown in Figure 1.  
The vibration of the \(m\)th lumped disk in the vertical \(y\) direction, which results from the vertical excitation force at the \(l\)th lumped element, can be computed using the flexibility matrix as follows:  
Figure 1: Distributed lumped model of rotor-bearing system. The vibration of the \(m\)th lumped disk in the vertical \(y\) direction, which results from the vertical excitation force at the \(l\)th lumped element, can be computed using the flexibility matrix as follows:  
\[G^{\prime}_{lm}(j\omega)=\underbrace{\Lambda_{rs}(j\omega)}_{r=4m-3,s=4l-2} \tag{1}\]  
where \(G^{\prime}_{lm}(j\omega)\) in (1) is the frequency response function, in which the corresponding flexibility matrix is as follows:  
\[\overset{\mathrm{n}}{\Lambda}= \begin{bmatrix}\begin{bmatrix}\mathbf{K_{l1}^{1} +B_{l1}^{1}}\end{bmatrix}&\mathbf{[K_{lr}^{1}]}&\mathbf{0}&\mathbf{0}& \mathbf{0}&\cdots&\mathbf{0}\\ \mathbf{[K_{rl}^{1}]}&\mathbf{[K_{rr}^{1}-K_{ll}^{2}+D_{rl}^{1}]}&-\begin{bmatrix} \mathbf{K_{lr}^{1}}\end{bmatrix}&\mathbf{0}&\cdots&\mathbf{0}\\ \mathbf{0}&\mathbf{[K_{rl}^{2}]}&\begin{bmatrix}\mathbf{K_{rr}^{2}-K_{ll}^{3}+ D_{rl}^{2}}\end{bmatrix}&-\begin{bmatrix}\mathbf{K_{lr}^{3}}\end{bmatrix}& \mathbf{0}&\cdots&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\begin{bmatrix}\mathbf{K_{rl}^{1}-1}\end{bmatrix}& \begin{bmatrix}\mathbf{K_{rr}^{1}-1}\end{bmatrix}&-\begin{bmatrix}\mathbf{K_{lr }^{1}-1}\end{bmatrix}&\cdots\mathbf{0}\\ \vdots&\vdots&\mathbf{0}&\mathbf{/}&\mathbf{/}&\mathbf{/}\mathbf{0}\\ 0&0&0&\cdots&\begin{bmatrix}\mathbf{K_{rl}^{n-1}}\end{bmatrix}&\begin{bmatrix} \mathbf{K_{rr}^{n-1}+B_{rl}^{n}}\end{bmatrix}\end{bmatrix} \]  
The matrices \(\mathbf{K}\) in the flexibility matrix are defined by the following:  
\[ \begin{bmatrix}\mathbf{K_{rr}^{i}}\end{bmatrix} = \begin{bmatrix}\mathbf{T_{rr}^ {i}}\end{bmatrix}\begin{bmatrix}\mathbf{T_{lr}^{i}}\end{bmatrix} ^{-1} \begin{bmatrix} \mathbf{K_{ll}^{i}}\end{bmatrix} =- \begin{bmatrix}\mathbf{T_{lr}^{i}}\end{bmatrix} ^{-1} \begin{bmatrix}\mathbf{T_{lr}^{i}}\end{bmatrix}\begin{bmatrix}\mathbf{K_{lr}^{i }}\end{bmatrix} =- \begin{bmatrix}\mathbf{T_{lr}^{i}}\end{bmatrix} ^{-1}\\ \begin{bmatrix}\mathbf{T_{lr}^{i}}\end{bmatrix} ^{-1} \begin{bmatrix} \mathbf{T_{lr}^{i}}\end{bmatrix} ^{-1} \begin{bmatrix}\mathbf{T_{lr}^{i}} \end{bmatrix} ^{-1} \begin{bmatrix}\mathbf{T_{lr}^{i}}\end{bmatrix}  \tag{2}\]  
where the \(\mathbf{T}\) matrices in (2) are as follows:  
\[ \begin{bmatrix}\mathbf{T_{lr}^{i}}\end{bmatrix} = \begin{bmatrix}\frac{\frac{ \cosh\sqrt{\Gamma}L+\cos\sqrt{\Gamma}L}{2}}{2\sqrt{\Gamma}}\begin{bmatrix} \sinh\sqrt{\Gamma}L+\sin\sqrt{\Gamma}L\\ \frac{\sqrt{\Gamma}}{2}\Big{(}\sinh\sqrt{\Gamma}L-\sin\sqrt{\Gamma}L\Big{)}& \frac{\cosh\sqrt{\Gamma}L+\cos\sqrt{\Gamma}L}{2}\end{bmatrix}&0&0\\ 0&0&\frac{\cosh\sqrt{\Gamma}L+\cos\sqrt{\Gamma}L}{2}&\frac{\sinh\sqrt{\Gamma}L +\sin\sqrt{\Gamma}L}{2\sqrt{\Gamma}}\\ 0&0&\frac{\sqrt{\Gamma}}{2}\Big{(}\sinh\sqrt{\Gamma}L-\sin\sqrt{\Gamma}L\Big{)}& \frac{\cosh\sqrt{\Gamma}L+\cos\sqrt{\Gamma}L}{2}\end{bmatrix} \end{bmatrix}\]  
\[ \begin{bmatrix}\mathbf{T_{lr}^{i}}\end{bmatrix} =