# Time-Domain Dynamic Modeling and Analysis of Complex Heavy-Duty Gearbox Considering Floating Effect
## 2 LTC Modeling Approach of Heavy-Duty gearbox
### Detailed Formulations
#### 2.2.4. System Dynamic Equations  
To construct the sun gear dynamic equations, supposing that \(k_{SB}\) and \(c_{SB}\) are the bearing stiffness and the rotational viscous damping coefficient of the sun gear bearing if it exists, respectively, and that there exist \(n\) planet gears:  
\[ \begin{split} M_{S}\ddot{x}_{S}+k_{SB}x_{S}-\sum\limits_{i=1}^{n} \left(N^{i}_{PS-x}+f^{i}_{PS-x}\right)=F_{Sx}\\ M_{S}\ddot{y}_{S}+k_{SB}y_{S}-\sum\limits_{i=1}^{n}\left(N^{i}_{ PS-y}+f^{i}_{PS-y}\right)=F_{Sy}+G_{S}\ \.\\ I_{S}\ddot{\theta}_{S}+c_{SB}\dot{\theta}_{S}-\sum\limits_{i=1}^{n }\left(T^{i}_{PS-N}+T^{i}_{PS-f}\right)=T_{S}\end{split}  \tag{31}\]  
\(F_{x}\), \(F_{y}\) and \(T\) in Equations (31)-(38) are the external forces and torque respectively, values of which are considered zeroes if they do not exist. Substituting the specific formulas of \(N^{i}_{PS}\), \(f^{i}_{PS}\) and \(T^{i}_{PS}\), the sun gear dynamic equations can be rewritten as  
\[ \begin{split} M_{S}\ddot{x}_{S}+k_{SB}x_{S}+\sum\limits_{i=1}^{n }\left\{N^{i}_{PS}\cdot\left[\sin(\gamma^{i}_{SP}-a^{i}_{SP})+\mu_{PS}\cos( \gamma^{i}_{SP}-a^{i}_{SP})\right]+c_{PS}\frac{d^{i}_{PS}(\mu_{PS}+\mu_{S})}{d^ {i}_{PS}}\cdot\dot{d}^{i}_{PS}\cos(\gamma^{i}_{SP}-a^{i}_{SP})\right\}=F_{Sx} \\ M_{S}\ddot{y}_{S}+k_{SB}y_{S}+\sum\limits_{i=1}^{n}\left\{N^{i}_{ PS}\cdot\left[\cos(\gamma^{i}_{SP}-a^{i}_{SP})+\mu_{PS}\sin(a^{i}_{SP}-\gamma^{i}_{ SP})\right]+c_{PS}\frac{d^{i}_{PS}(\mu_{PS}+\mu_{S})}{d^{i}_{PS}}\dot{d}^{i}_{PS} \cos(a^{i}_{SP}-\gamma^{i}_{SP})\right\}=F_{Sy}\ \.\\ I_{S}\ddot{\theta}_{S}+c_{SB}\dot{\theta}_{S}+\sum\limits_{i=1}^{n }\left\{N^{i}_{PS}\cdot\left[\tau_{Sb}-\mu_{PS}\frac{Z_{S}}{Z_{S}+Z_{S}}d^{i}_ {PS}\sin a^{i}_{PS}\right]-c_{PS}\frac{Z_{SB}(\mu_{PS}+\mu_{S})}{(Z_{S}+Z_{S}) \mu_{PS}}\sin a^{i}_{PS}\cdot\dot{d}^{i}_{PS}\right\}=T_{S}\end{split}  \tag{32}\]For the ring gear, the dynamic equations derivation process is similar. \(k_{RB}\) and \(c_{RB}\) are assumed to be the ring bearing stiffness and the rotational viscous damping coefficient if they exist, respectively. \[ \begin{array}{l}M_{R}\ddot{x}_{R}+k_{RB}x_{R}-\sum\limits_{i=1}^{n}\left(N_{PR -x}^{i}+f_{PR-x}^{i}\right)=F_{Rx}\\ M_{R}\ddot{y}_{R}+k_{RB}y_{R}-\sum\limits_{i=1}^{n}\left(N_{PR-y}^{i}+f_{PR-y}^{ i}\right)=F_{Ry}+G_{R}\\ I_{R}\ddot{\theta}_{R}+c_{RB}\dot{\theta}_{R}-\sum\limits_{i=1}^{n}\left(T_{PR- N}^{i}+T_{PR-f}^{i}\right)=T_{R}\end{array} . \tag{33}\]  
Plugging in the corresponding terms, the equations can be rewritten as  
\[ \begin{array}{l}M_{R}\ddot{x}_{R}+k_{RB}x_{R}+\sum\limits_{i=1}^{n}\left\{N_ {PR}^{i}\times\left[\sin(\gamma_{RP}^{i}+\dot{a}_{RP}^{i})+\mu_{PR}\cos(\gamma _{RP}^{i}+\dot{a}_{RP}^{i})\right]+c_{PR}\frac{d_{\rm{PB}}(r_{R}+r_{\rm{R}})}{ d_{R}^{2}}\dot{d}_{PR}^{i}\cos(\gamma_{RP}^{i}+\dot{a}_{RP}^{i})\right\}=F_{Rx}\\ M_{R}\ddot{y}_{R}+k_{RB}y_{R}+\sum\limits_{i=1}^{n}\left\{N_{PR}^{i}\times\left[ \cos(\gamma_{RP}^{i}+\dot{a}_{RP}^{i})+\mu_{PR}\sin(\alpha_{RP}^{i}+\gamma_{RP }^{i})\right]+c_{PR}\frac{d_{\rm{PB}}(r_{R}+r_{\rm{R}})}{d_{R}^{2}}\dot{d}_{PR} ^{i}\cos(\alpha_{RP}^{i}-\gamma_{RP}^{i})\right\}=F_{Ry}+G_{R}\\ I_{R}\ddot{\theta}_{R}+c_{SB}\dot{\theta}_{S}+\sum\limits_{i=1}^{n}\left\{N_{ PR}^{i}\times\left[r_{Rb}+\mu_{PR}\frac{Z_{\rm{S}}}{Z_{R}+Z_{R}}d_{PR}^{i}\sin \dot{a}_{PR}^{i}\right]-c_{PR}\frac{Z_{\rm{S}}d_{\rm{PB}}(r_{R}+r_{\rm{R}})}{ (Z_{R}+Z_{R})\dot{\theta}_{R}}\sin\alpha_{PR}^{i}\times\dot{d}_{PR}^{i}\right\} =T_{R}\end{array}  \tag{34}\]  
For the \(i\)th planet gear, as the applied forces on it include the engaging forces from the ring and sun gear and the bearing force, its dynamic equations are given by  
\[