## 2 Deterministic Model of a Dual-Rotor System  
### Calculation Model of a Dual-Rotor System  
This paper cites the calculation model from Reference [21], and a structure diagram of a dual-rotor system with a single intermediary support is shown in Figure 1. The system is divided into 10 nodes according to the concentrated mass rigid disk, massless elastic shaft segment, and support. It consists of two rotors: rotor \(I\) is an inner rotor, and rotor \(II\) is an outer rotor. Rotor \(I\) is connected to the base by the first and sixth nodes, while the left end of rotor \(II\) is connected to the base by the seventh node, and the right end is connected to rotor I by the tenth node. The detailed parameters are provided in Table 1.  
From Figure 1, the number of nodes in rotor \(I\) and \(II\) are unequal, which is counter to constructing a whole transfer matrix. Nevertheless, virtual elements with zero mass and zero length are added at both end sections of the dual-rotor system, and the number of elements is reset [22]. The two rotors are also elastomers; the 2 nd, 3 rd, 5 th, 6 th, 7 th, 8 th, 10 th, and 11 th elements are elastomers, and the 4 th and 9 th elements are rigid bodies. Therefore, the elastic deformation will also have an impact on the dynamic characteristics of the system, which must be taken into account [23; 24; 25]. This problem will be studied in the following section. The dual-rotor system, after resetting, is shown in Figure 2. All the derivation processes described later in this paper are based on it. The state vector of the \(i\)th node of rotors \(I\) and \(II\) can be written as \(\mathbf{Z}_{i}^{I}=[M^{I},Q^{I},x^{I},\theta]_{i}^{T}\) and \(\mathbf{Z}_{i}^{II}=[M^{II},Q^{II},x^{II},\theta]_{i}^{T}\), where the parameters \(M,Q,x,\) and \(\theta\) are the bending moment, shear force, displacement, and rotation angle, respectively. To facilitate the construction of the transfer model in the overall elements, \(\mathbf{Z}_{i}^{I}\) and \(\mathbf{Z}_{i}^{II}\) are combined in the state vector, \(\mathbf{Z}_{i}\). \[\mathbf{Z}_{i}= \begin{bmatrix}\mathbf{Z}_{i}^{I}\\ \mathbf{Z}_{i}^{II}\end{bmatrix}  \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Rotor**} & \multirow{2}{*}{\begin{tabular}{c} **Speed** \\ **\(\omega\)/(rad\(\cdot\)s\({}^{-1}\))** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Stiffness of** \\ **Rotor** \\ **\(J\)/(m\({}^{4}\))** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Mass** \\ **\(m\)/kg** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Moment of** \\ **Inertia** \\ **\(I_{p}\)/(kg\(\cdot\)m\({}^{2}\))** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Stiffness of** \\ **Element** \\ \(k\times 10^{-7}\)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Length** \\ **\(I\)/(m)** \\ \end{tabular} } \\ \cline{5-8}  & & & 1 & 0.05770 & 0 & 0.6269 & 0.0762 \\  & & & 2 & 10.7023 & 0.0859 & & 0.1778 \\ \multirow{2}{*}{\(I\)} & \multirow{2}{*}{1047.2} & \multirow{2}{*}{2.6467 \(\times\)\(10^{-9}\)} & \multirow{2}{*}{3} & \multirow{2}{*}{0.24990} & \multirow{2}{*}{0} & \multirow{2}{*}{0.8756} & \multirow{2}{*}{0.0508} \\  & & & 5 & 7.08690 & 0.0678 & & 0.0508 \\ \multirow{2}{*}{\(I\)} & \multirow{2}{*}{1570.8} & \multirow{2}{*}{2.1935 \(\times\)\(10^{-3}\)} & \multirow{2}{*}{6} & \multirow{2}{*}{0.04699} & \multirow{2}{*}{0} & \multirow{2}{*}{1.7513} & \multirow{2}{*}{0.0508} \\  & & & & 8 & 7.20200 & 0.0429 & & 0.1524 \\ \multirow{2}{*}{\(II\)} & \multirow{2}{*}{1570.8} & \multirow{2}{*}{2.1935 \(\times\)\(10^{-3}\)} & \multirow{2}{*}{9} & \multirow{2}{*}{3.69200} & \multirow{2}{*}{0.0271} & \multirow{2}{*}{0.0276} & \multirow{2}{*}{0.0508} \\  & & & & 10 & 0.04699 & 0 & 0.8756 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the dual-rotor system. Figure 2: Element division of the dual-rotor system. Since this system is a complex rotor system composed of two rotors, the traditional transfer matrix method for a single-rotor system is no longer applicable.