# Vibration Properties of Dual-Rotor Systems under Base Excitation, Mass Unbalance and Gravity
## 2 Modeling for a Dual-Rotor System under Base Motions
### EOMs for the Shaft
For a circular shaft, the \(i_{p}=2i_{d}\) and \(i_{d}=\rho I_{s}\) where \(l_{s}=\pi\big{(}r_{so}^{4}-r_{si}^{4}\big{)}/4\) (\(r_{so}\) is the shaft outer radius and \(r_{si}\) is the shaft inner radius). The strain energy of a shaft element is given by  
\[U_{sb}=\frac{1}{2}\int_{0}^{l}EI_{s}\Bigg{[}\bigg{(}\frac{\partial\theta_{s}}{ \partial y}\bigg{)}^{2}+\bigg{(}\frac{\partial\psi_{s}}{\partial y}\bigg{)}^{2 }\Bigg{]}\mathrm{d}y+\frac{1}{2}\int_{0}^{l}\kappa GA_{s}\Bigg{[}\bigg{(}\frac{ \partial x_{s}}{\partial y}+\psi_{s}\bigg{)}^{2}+\bigg{(}\frac{\partial z_{s}}{ \partial y}-\theta_{s}\bigg{)}^{2}\Bigg{]}\mathrm{d}y \tag{25}\]  
where \(E\) is the Young's modulus of the shaft material. Let \(z_{s0}\) be the component of the absolute position vector \(\left(\mathbf{O_{0}C_{s}}\right)_{F_{0}}\) along the axis \(O_{0}Z_{0}\). The gravitational potential energy of a shaft element is given by  
\[U_{sg}=\rho_{s}A_{s}g\int_{0}^{l}z_{s0}\mathrm{d}y \tag{26}\]  
After substituting \(T_{s}\), \(U_{sb}\) and \(U_{sg}\) into the Lagrange equation, simplifying (\(\sin\theta_{s}\approx\theta_{s}\), \(\sin\psi_{s}\approx\psi_{s}\), \(\cos\theta_{s}\approx 1\)) and ignoring the high-order terms, the EOMs for a shaft element was accomplished as  
\[\mathbf{m_{s}\ddot{q}_{s}}+\left(\Omega_{s}\mathbf{g_{s0}}+\mathbf{c_{s}} \right)\dot{\mathbf{q}_{s}}+\left(\mathbf{k_{s0}}+\mathbf{k_{s}}\right) \mathbf{q_{s}}=\mathbf{f_{str}}+\mathbf{f_{slb}}+\mathbf{f_{san}}+\mathbf{f_{ sg}} \tag{27}\]  
where \(\mathbf{m_{s}}\), \(\mathbf{g_{s0}}\) and \(\mathbf{k_{s0}}\) are the classical mass, gyroscopic and stiffness matrices of the Timoshenko beam element, respectively; \(\mathbf{c_{s}}\) and \(\mathbf{k_{s}}\) are additional time-varying damping and stiffness matrices due to the time-variable base rotations; \(\mathbf{f_{str}}\), \(\mathbf{f_{san}}\) and \(\mathbf{f_{sg}}\) are external forces motivated by base translations, base rotations and gravity, respectively; and \(\mathbf{f_{slb}}\) are additional external forces due to both the position vector \(\mathbf{O_{b}O_{brg1}}\) in Figure 2 and base rotations. The origin of the base frame is set right in the center of bearing 1; thus, \(\mathbf{f_{slb}}=\mathbf{0}\). The precise coefficient matrices and force vectors are listed as follows:  
\[\mathbf{m_{s}}=\int_{0}^{l}\rho A_{s}\psi(y)^{\mathsf{T}}\psi(y)\mathrm{d}y+ i_{d}\int_{0}^{l}\mathbf{\Phi}(y)^{\mathsf{T}}\mathbf{\Phi}(y)\mathrm{d}y \tag{28}\]  
\[\mathbf{g_{s0}}=i_{p}\int_{0}^{l}\mathbf{\Phi}(y)^{\mathsf{T}}\left[ \begin{array} []{cc}0&-1\\ 1&0\end{array} \right]\mathbf{\Phi}(y)\mathrm{d}y \tag{29}\]  
\[\mathbf{\Gamma}(y)=\mathbf{\[PERSON]}^{(y)}+\left[ \begin{array}{cc}0&1\\ -1&0\end{array} \right]\mathbf{\[PERSON]}(y) \tag{30}\]  
\[\mathbf{k_{s0}}=\int_{0}^{l}EI_{s}\mathbf{\[PERSON]}(y)^{\mathsf{T}}\mathbf{\[PERSON]}( y)\mathrm{d}y+\int_{0}^{l}\kappa GA_{s}\mathbf{\Gamma}(y)^{\mathsf{T}} \mathbf{\Gamma}(y)\mathrm{d}y \tag{31}\]  
\[\mathbf{c_{s}}=\left(2\int_{0}^{l}\rho A_{s}\psi(y)^{\mathsf{T}}\left[ \begin{array} []{cc}0&1\\ -1&0\end{array} \right]\mathbf{\Psi}(y)\mathrm{d}y+\left(i_{p}-2i_{d}\right) \int_{0}^{l}\mathbf{\Phi}(y)^{\mathsf{T}}\left[ \begin{array}{cc}0&-1\\ 1&0\end{array} \right]\mathbf{\Phi}(y)\mathrm{d}y\right)\omega_{by} \tag{32}\]  
\[\mathbf{k_{s}}=\mathbf{k_{s1}}\dot{\omega}_{by}+\mathbf{k_{s2}}\omega_{bx}^{2 }+\mathbf{k_{s3}}\omega_{by}^{2}+\mathbf{k_{s4}}\omega_{bz}^{2}+\mathbf{k_{s5 }}\omega_{bx}\omega_{bz}+\mathbf{k_{s6}}\Omega_{r}\omega_{by} \tag{33}\]  
\[\mathbf{k_{s1}}=\int_{0}^{l}\rho A_{s}\psi(y)^{\mathsf{T}}\left[ \begin{array}[ ]{cc}0&1\\ -1&0\end{array} \right]\mathbf{\Psi}(y)\mathrm{d}y+\int_{0}^{l}\mathbf{\Phi}(y)^ {\mathsf{T}}\left[ \begin{array}{cc}0&i_{d}\\ i_{p}-i_{d}&0\end{array} \right]\mathbf{\[PERSON]}(y)\mathrm{d}y \tag{34}\]  
\[\mathbf{k_{s2}}=\int_{0}^{l}\rho A_{s}\psi(y)^{\mathsf{T}}\left[ \begin{array}[ ]{cc}0&0\\ 0&-1\end{array} \right]\mathbf{\[PERSON]}(y)\mathrm{d}y+\left(i_{p}-i_{d}\right)\int_{ 0}^{l}\mathbf{\[PERSON]}(y)^{\mathsf{T}}\left[