## 2 Deterministic Model of a Dual-Rotor System
### The Riccati Whole Mean Transfer Model of the Uncoupled Element  
Reference [26] points out that a maximum entropy random matrix can be constructed by combining RMT and MEP. Nevertheless, the whole mean transfer matrix has to be known in this process. As a consequence, this Sections 2.2 and 2.3 will focus on how to construct the Riccati whole mean transfer matrix of any element. Meanwhile, the construction method of the maximum entropy random matrix will be provided in detail in Sections 3.1 and 3.2. There are no intermediary supports in the uncoupled elements. Consequently, the transfer matrices need not consider the action of the coupling matrix. That is, the coupling term in the whole mean transfer matrices is zero. Therefore, the \(\underline{\mathbf{U}}_{i}\) can be expressed as  
\[\underline{\mathbf{U}}_{i}=\left[ \begin{array}{cc}\underline{\mathbf{U}}^{I}&\mathbf{0 }\\ \mathbf{0}&\underline{\mathbf{U}}^{II}\end{array} \right]_{i} \tag{3}\]  
where \(\mathbf{U}^{j}_{i}\) can be written as  
\[\underline{\mathbf{U}}^{j}_{i}=\left[ \begin{array}{ccc}1&l^{j}&\left(m^{j} \omega^{2}-k_{s}^{j}\right)l^{j}&\left(l_{P}^{j}-l_{d}^{j}\frac{\Omega^{j}}{ \omega}\right)\omega^{2}\\ 0&1&m^{j}\omega^{2}-k_{s}^{j}&0\\ \frac{(l^{j})^{2}}{2(EI)^{j}}&\frac{(l^{j})^{3}(1-\Gamma)}{6(EI)^{j}}&1+\frac {\left(m^{j}\omega^{2}-k_{s}^{j}\right)(l^{j})^{3}(1-\Gamma)}{6(EI)^{j}}&l^{j }+\frac{\left(l_{P}^{j}-l_{d}^{j}\frac{\Omega^{j}}{\omega}\right)(l^{j})^{2} \omega^{2}}{2(EI)^{j}}\\ \frac{(l^{j})}{(EI)^{j}}&\frac{(l^{j})^{2}}{2(EI)^{j}}&\frac{\left(m^{j}\omega ^{2}-k_{s}^{j}\right)(l)^{2}}{2(EI)^{j}}&1+\frac{\left(l_{P}^{j}-l_{d}^{j} \frac{\Omega^{j}}{\omega}\right)l^{j}\omega^{2}}{(EI)^{j}}\end{array} \right]_{i} \tag{4}\]  
where \(m^{j}_{i}\), \(l^{j}\), \(E^{j}_{i}\), \(l^{j}_{i}\), and \(\Omega^{j}_{i}\) represent the concentrated mass, shaft segment length, elastic modulus, section moment of inertia, and rotation angular velocity of the \(j\)th rotor at the \(i\)th element, respectively; \(J_{p}\)\({}^{j}_{i}\) and \(J_{d}\)\({}^{j}_{i}\) are the polar moment of inertia and the diametric moment of inertia at the rigid thin disk, respectively; \(k_{s}\)\({}^{j}_{j}\) is the total stiffness of the connection between the shaft segment and base; \(\omega\) is the angular velocity of precession; \(\Gamma\) = 6 _E1)/(ktGAl2)_ is the shear influence coefficient; \(G\) and \(A\) are the material shear modulus and cross-section area; and \(k_{t}\) is the cross-section coefficient (0.886 for solid shafts and 2/3 for thin-walled hollow shafts). From Equations (3) and (4), the whole mean transfer matrices of the uncoupled elements can be derived. Yet, with a large number of nodes in the system, a computer value overflow will occur; furthermore, it may appear that a large number subtracts another large number in calculating the vibration modes, which will decrease the calculation accuracy and result in the vibration mode distortion [27]. To avoid the above matters, the Riccatiwhole mean transfer matrices, \(\textbf{S}_{i}\), of the uncoupled elements are constructed by using the method in Reference [21], and the calculation formula is as follows:  
\[ \begin{split}\textbf{S}_{i+1}=\bigg{(}\begin{bmatrix}\underline{ \textbf{U}}_{11}^{I}&\textbf{0}\\ \textbf{0}&\underline{\textbf{U}}_{11}^{II}\end{bmatrix}\textbf{S}_{i}+ \begin{bmatrix}\underline{\textbf{U}}_{12}^{I}&\textbf{0}\\ \textbf{0}&\underline{\textbf{U}}_{12}^{II}\end{bmatrix}_{i}\bigg{)}\cdot\\ \bigg{(}\begin{bmatrix}\underline{\textbf{U}}_{21}^{I}&\textbf{0}\\ \textbf{0}&\underline{\textbf{U}}_{21}^{II}\end{bmatrix}_{i}\textbf{S}_{i}+ \begin{bmatrix}\underline{\textbf{U}}_{22}^{I}&\textbf{0}\\ \textbf{0}&\underline{\textbf{U}}_{22}^{II}\end{bmatrix}_{i}\bigg{)}^{-1} \end{split}