# Time-Domain Dynamic Modeling and Analysis of Complex Heavy-Duty Gearbox Considering Floating Effect
## 2 LTC Modeling Approach of Heavy-Duty gearbox
### Detailed Formulations
#### 2.2.2 Force Derivation of the Engagement of the Ring and \(i\)th Planet Gear  
Analysis procedures to derive the force between the ring and \(i\)th planet gears are similar to the analysis procedures for that between the sun and \(i\)th planet gears. The difference is mainly concentrated in a small fluctuation in meshing stiffness between the ring and planet gear due to the high contact ratio, such that meshing stiffness can be regarded as a constant in some situations, which will be proved via parameter identification results in a later section. The center distance of the \(i\)th planet and ring gear is  
\[d_{PR}^{i}=\sqrt{\left(x_{P}^{i}-x_{R}\right)^{2}+\left(y_{P}^{i}-y_{R}\right) ^{2}}. \tag{16}\]  
The time-varying position angle between the center connection line and the X axis is  
\[\gamma_{PR}^{i}=\arctan\left[\left(y_{P}^{i}-y_{R}\right)/\left(x_{P}^{i}-x_{ R}\right)\right]\ \&\ \Delta\gamma_{PR}^{i}=\gamma_{PR}^{i}-\gamma_{PR0}^{i}. \tag{17}\]  
Under the ideal condition, the rotational angles fit a transmission ratio equation as follows:  
\[\left(\theta_{P}^{i}-\Delta\gamma_{PR}^{i}\right)\cdot Z_{P}-\left(\theta_{R }-\Delta\gamma_{PR}^{i}\right)\cdot Z_{R}=0. \tag{18}\]  
The ideal angle \(\theta_{P-PR}^{i}\) of a planet gear under the condition of rotation angle of \(\theta_{R}\) and \(\Delta\gamma_{PR}^{i}\) is  
\[\theta_{P-PR}^{i}=\left(\theta_{R}-\Delta\gamma_{PR}^{i}\right)\cdot Z_{R}/Z_ {P}+\Delta\gamma_{PR}^{i}. \tag{19}\]  
As a result, the angle difference \(\Delta\theta_{P-PR}^{i}\) with respect to the \(i\)th planet gear is  
\[\Delta\theta_{P-PR}^{i}=\theta_{P}^{i}-\theta_{P-PR}^{i}. \tag{20}\]  
The linear deformation caused by the angle difference is  
\[\delta_{\theta_{m-PR}}^{i}=-r_{P-PR}^{i}\cdot\tan\Delta\theta_{P-PR}^{i}\cdot \cos\alpha_{PR}^{i}, \tag{21}\]  
where \(\alpha_{PR}^{i}\) is the engagement angle and \(r_{P-PR}^{i}\) is the time-varying pitch radius of the \(i\)th planet gear, which can be derived from the center distance \(d_{PR}^{i}\), \(r_{P-PR}^{i}=d_{PR}^{i}\cdot Z_{P}/\left(Z_{R}-Z_{P}\right)\). The linear deformation caused by the variation in center distance is  
\[\delta_{dt-PR}^{i}=\Delta d_{PR}^{i}\cdot\sin\alpha_{PR}^{i}. \tag{22}\]  
Thus, the equivalent normal composite deformation of the spring damper between the ring and \(i\)th planet gears is  
\[\delta_{n-PR}^{i}=\delta_{\theta n-PR}^{i}+\delta_{dt-PR}^{i}=-r_{P-PR}^{i} \cdot\tan\Delta\theta_{P-PR}^{i}\cdot\cos\alpha_{PR}^{i}+\Delta d_{PR}^{i} \cdot\sin\alpha_{PR}^{i}. \tag{23}\]  
As in the previous section, the tangent deformation mainly caused by variation in center distance between the ring and planet gear is  
\[\delta_{t-PR}^{i}=-\Delta d_{PR}^{i}\cdot\cos\alpha_{PR}^{i}. \tag{24}\]  
\(k_{PR}\) denotes the equivalent meshing stiffness of the ring and planet gears, and the fluctuation its value can be ignored. Based on the previous derivation of the deformation, the normal force on the \(i\)th planet gear can be computed as  
\[ \begin{array}{l}N_{RP}^{i}=k_{PR}\cdot\delta_{n-PR}^{i}\\ =k_{PR}\cdot\left[-\frac{Z_{P}\left(r_{R-PR}\right)}{Z_{R}-Z_{P}}\tan(\theta_ {P}^{i}-\frac{Z_{R}}{Z_{P}}\theta_{R}^{i}+\frac{Z_{R}-Z_{P}}{Z_{P}}\Delta \gamma_{PR}^{i})+\left(d_{PR}^{i}-d_{PR0}^{i}\right)\cdot\sin\alpha_{PR}^{i} \right]\,\end{array}  \tag{25}\]Its direction is tangent to the base circle of the \(i\)th planet gear, and the angle between it and the X axis is  
\[\beta^{i}_{RP-N}=\pi/2+\gamma^{i}_{PR}+\alpha^{i}_{PR}. \tag{26}\]  
The tangent friction force \(f^{i}_{RP}\) between the \(i\)th planet gear and ring gear is composed of the viscous damping and coulomb friction forces:  
\[f^{i}_{RP}=(\mu_{PR}N^{i}_{RP}+c_{PR}\cdot\dot{\delta}^{i}_{i-PR})\cdot\text{ sgn}(\dot{\delta}^{i}_{i-PR}), \tag{27}\]  
in which the definitions of \(\mu_{PR}\), \(c_{PR}\) and the function \(\text{sgn}(x)\) are almost the same as in the previous subsection. The tangent friction force is perpendicular to the normal force, direction of which is opposite to the direction of relative sliding velocity.