# Vibration Properties of Dual-Rotor Systems under Base Excitation, Mass Unbalance and Gravity
## 3 Results and Discussions
### Computation Method and Validation  
From the results in Section 2.3.2, it is apparent that the mass, gyroscopic, damping and stiffing matrices on the left of the EOMs are constant and the forces on the right of the EOMs are constant or harmonic. Thus, the natural properties and responses of the linear dual-rotor system can be solved by direct matrix operations in the complex field. Introducing a state-space vector \(\mathbf{h}=\left[ \begin{array}{c}\mathbf{q}\\ \dot{\mathbf{q}}\end{array} \right]\) and assuming that \(\mathbf{q}=\Phi e^{\lambda t}\), the second-order homogeneous EOMs for the dual-rotor system with constant base rotations can be transformed into first-order state space form  
\[\left[ \begin{array}{cc}\mathbf{K}_{0}+\mathbf{K}_{\mathrm{b}}&\mathbf{0}\\ \mathbf{0}&\mathbf{M}\end{array} \right]\dot{\mathbf{h}}+\left[ \begin{array}[] {cc}\mathbf{0}&-(\mathbf{K}_{0}+\mathbf{K}_{\mathrm{b}})\\ \mathbf{K}_{0}+\mathbf{K}_{\mathrm{b}}&\Omega_{LP}\mathbf{G}_{0,LP}+\Omega_{HP }\mathbf{G}_{0,HP}+\mathbf{C}_{\mathrm{b}}\end{array} \right]\mathbf{h}= \mathbf{0} \tag{71}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Rotors** & **Node–Node** & \(\mathbf{I}\) **(mm)** & \(\boldsymbol{r_{si}}\) **(mm)** & \(\boldsymbol{r_{so}}\) **(mm)** \\ \hline LP & 1–2, 2–3 & 75 & 0 & 6 \\ LP & 3–4 & 180 & 0 & 6 \\ LP & 4–5, 5–6 & 120 & 0 & 6 \\ LP & 6–7, 7–8 & 50 & 0 & 6 \\ HP & 9–10, 10–11 & 120 & 19.9 & 27.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Shaft parameters for the dual-rotor-bearing system. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Rotors** & **Node** & \(\boldsymbol{k_{b}}\) **(kg\(\cdot\)m\({}^{3}\))** & \(\boldsymbol{c_{b}}\) **(N\(\cdot\)m\({}^{-1}\))** \\ \hline
1 & 1 & \(5\times 10^{7}\) & 100 \\
2 & 2 & \(5\times 10^{7}\) & 100 \\
3 & 6, 9 & \(5\times 10^{7}\) & 100 \\
4 & 8 & \(5\times 10^{7}\) & 100 \\
5 & 11 & \(5\times 10^{7}\) & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Bearing parameters for the dual-rotor-bearing system. Figure 3: The finite element model for the dual-rotor-bearing system: 1–11 are the serial numbers of nodes, 1⃝1⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝5⃝�5⃝5⃝5⃝5⃝�5⃝5⃝�5⃝5⃝5⃝�5⃝5⃝�5⃝5⃝�5⃝�5⃝�5⃝�5⃝�5⃝�5⃝�5⃝�5⃝�5⃝�5⃝�5⃝�5⃝�5⃝�5⃝��5⃝�5⃝��5⃝��5⃝��5⃝��5⃝��5⃝��5⃝��5⃝��5⃝���5⃝���5⃝���5⃝���5❝����5❝����5❝�����5❝����5❝�����5❝�����5❝������5❝�����5❝������5❝������5❝������5❝�������5❝�������5❝��������5❝�������5❝��������5❝������5�The equivalent eigenvalue problem is  
\[-\left[ \begin{array}{cc}\mathbf{K}_{0}+\mathbf{K}_{\mathrm{b}}&\mathbf{0}\\ \mathbf{0}&\mathbf{M}\end{array} \right]^{\mathrm{T}}\!\!\left[ \begin{array}{ cc}\mathbf{0}&-(\mathbf{K}_{0}+\mathbf{K}_{\mathrm{b}})\\ \mathbf{K}_{0}+\mathbf{K}_{\mathrm{b}}&\Omega_{LP}\mathbf{G}_{0,\mathrm{LP}}+ \Omega_{HP}\mathbf{G}_{0,\mathrm{HP}}+\mathbf{C}_{\mathrm{b}}\end{array}  \right]\left[ \begin{array}{cc}\Phi\\ \lambda\Phi\end{array} \right]=\lambda\left[ \begin{array}{cc}\Phi\\ \lambda\Phi\end{array} \right] \tag{72}\]  
The solved \(\lambda\) and \(\Phi\) are the natural frequency and corresponding mode shape vector. For the constant forces assumed as \(\mathbf{F}_{0}\), the responses are solved by  
\[\mathbf{q}_{0}=[\mathbf{K}_{0}+\mathbf{K}_{\mathrm{b}}]^{T}\mathbf{F}_{0} \tag{73}\]  
The harmonic forces assumed as \(\mathbf{F}_{\mathrm{v}}=\mathbf{V}_{\mathrm{c}}\cos(\omega t)+\mathbf{V}_{ \mathrm{s}}\sin(\omega t)\) can be rewritten in a complex field as  
\[\mathbf{F}_{\mathrm{v}}=(\mathbf{V}_{\mathrm{c}}-j\mathbf{V}_{\mathrm{s}})e^ {j\omega t} \tag{74}\]  
where \(j\) is the imaginary unit.