# Vibration Properties of Dual-Rotor Systems under Base Excitation, Mass Unbalance and Gravity
## 2 Modeling for a Dual-Rotor System under Base Motions
### FEM Equations of the Dual-Rotor System  
#### 2.3.1 EOMs for Dual-Rotor System under General Base Motions  
The number of beam elements for the two shafts is denoted as \(n_{e}\), and the number of nodes is \(n_{e}+2\). After assembling equations for the disks, mass unbalance, linear bearings and shaft elements, the finite element equations for the dual-rotor system with base motions are obtained:  
\[\mathbf{M}\mathbf{\tilde{q}}+(\Omega_{LP}\mathbf{G}_{0,LP}+\Omega_{HP}\mathbf{ G}_{0,HP}+\mathbf{C}_{0}+\mathbf{C}_{b})\mathbf{\dot{q}}+(\mathbf{K}_{0}+ \mathbf{K}_{b})\mathbf{q}=\mathbf{F}_{\text{btr}}+\mathbf{F}_{\text{blb}}+ \mathbf{F}_{\text{ban}}+\mathbf{F}_{u}+\mathbf{F}_{g} \tag{50}\]  
where \(\mathbf{q}\) is the \(4(n_{e}+2)\times 1\) dimension displacement vector. \(\mathbf{M}=\sum\left(\mathbf{m}_{d}+\mathbf{m}_{s}\right)\); \(\mathbf{G}_{0}=\sum\left(\mathbf{g}_{d0}+\mathbf{g}_{s0}\right)\); \(\mathbf{C}_{0}=\sum\left(\mathbf{\tilde{q}}\mathbf{M}+\eta\mathbf{k}_{s0}+ \mathbf{c}_{\text{prog}}\right)\); and \(\mathbf{K}_{0}=\sum\left(\mathbf{k}_{s0}+\mathbf{k}_{\text{bbrg}}\right)\). These are, respectively, the conventional mass, gyroscopic, damping and stiffness matrices, and \(\sum\) represents the assembly of EOMs of the sub-systems. \(\mathbf{C}_{b}\) and \(\mathbf{K}_{b}\) are additional time-variable damping and stiffness matrices due to base angular motions. The subscripts 'tr', 'lb', 'an', 'u' and 'g' on the right of the equation represent the sources of the external forces; namely, base translations, left bearing, base angular motions, mass unbalance and gravity. Part of these matrices and vectors are given by  
\[\mathbf{C}_{b}=\mathbf{C}_{b1}\omega_{by} \tag{51}\]  
\[\mathbf{K}_{b}=\mathbf{K}_{b1}\dot{\omega}_{by}+\mathbf{K}_{b2}\omega_{bx}^{2}+ \mathbf{K}_{b3}\omega_{by}^{2}+\mathbf{K}_{b4}\omega_{bz}^{2}+\mathbf{K}_{b5 }\omega_{bx}\omega_{bz}+(\mathbf{K}_{b6,LP}\Omega_{LP}+\mathbf{K}_{b6,HP} \Omega_{HP})\omega_{by} \tag{52}\]  
\[\mathbf{F}_{\text{btr}}(t)=-\mathbf{p}_{\text{b1}}a_{bx}-\mathbf{p}_{\text{b2}}a _{bz} \tag{53}\]  
\[\mathbf{F}_{\text{blb}}(t)=\mathbf{p}_{\text{b1}}\Big{(}x_{lb}(\omega_{by}^{2}+ \omega_{bz}^{2})+y_{lb}(\dot{\omega}_{bz}-\omega_{bx}\omega_{by})-z_{lb}(\dot{ \omega}_{by}+\omega_{bx}\omega_{bz})\Big{)} \tag{54}\]  
\[+\mathbf{p}_{\text{b2}}\Big{(}x_{lb}(\dot{\omega}_{by}-\omega_{bx}\omega_{bz})-y_ {lb}(\dot{\omega}_{bx}+\omega_{by}\omega_{bz})+z_{lb}(\omega_{bx}^{2}+\omega_{by }^{2})\Big{)}\]  
\[\mathbf{F}_{\text{ban}}(t)=\mathbf{s}_{\text{b1}}\dot{\omega}_{bx}+\mathbf{s}_{ \text{b2}}\dot{\omega}_{bz}+\mathbf{s}_{\text{sb3}}\omega_{bx}\omega_{by}+ \mathbf{s}_{\text{b4}}\omega_{by}\omega_{bz}+(\mathbf{s}_{\text{b5,LP}}\Omega_{LP}+ \mathbf{s}_{\text{b5,HP}}\Omega_{HP})\omega_{bx} \tag{55}\]  
\[+(\mathbf{s}_{\text{b6,LP}}\Omega_{LP}+\mathbf{s}_{\text{b6,HP}}\Omega_{HP}) \omega_{bz} \tag{56}\]  
\[\mathbf{F}_{g}(t)=\mathbf{p}_{\text{b1}}g\cos a\sin\beta-\mathbf{p}_{\text{b2 }}g\cos a\cos\beta \tag{57}\]  
where \(\mathbf{C}_{b1}=\sum\left(\mathbf{c}_{\text{d1}}+\mathbf{c}_{\text{s1}}\right)\); \(\mathbf{K}_{\text{bi}}=\sum\left(\mathbf{k}_{\text{di}}+\mathbf{k}_{\text{si}}\right)\); \(\mathbf{p}_{\text{bj}}=\sum\left(\mathbf{p}_{\text{dj}}+\mathbf{p}_{\text{ sj}}\right)\); \(\mathbf{s}_{\text{bi}}=\sum\left(\mathbf{s}_{\text{di}}+\mathbf{s}_{\text{si}}\right)\); and (\(\text{i}=1,2,\cdots 6,\,\text{j}=1,2\)). Those vectors and matrices--including \(\mathbf{C}_{b1}\), \(\mathbf{K}_{\text{bi}}\), \(\mathbf{p}_{\text{bj}}\) and \(\mathbf{s}_{\text{bi}}\)--are determined by geometric and material parameters and kept constant for a dual-rotor system.