# Time-Domain Dynamic Modeling and Analysis of Complex Heavy-Duty Gearbox Considering Floating Effect
## 2 LTC Modeling Approach of Heavy-Duty gearbox
### Detailed Formulations
#### 2.2.1 Force Derivation of the Engagement of the Sun and \(i\)th Planet Gears  
The composite deformation in normal and tangent directions of a meshing can be obtained from the generalized displacements; thus, the normal force \(N\) and tangent force \(f\) are derived. As shown in Figure 3, two main influencing factors exist: the difference between the actual and the ideal angle, and the time-varying center distance between the sun and planet gear that is mainly introduced by the floating components. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Existing Method** & **Proposed Method** \\ \hline Frequency-domain behavior & \(\surd\) & \(\surd\) \\ Transient time-domain behavior & \(\surd\) & \(\surd\) \\ Continuous time-domain behavior & \(\times\) & \(\surd\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of existing and proposed method. Figure 2: Force analysis of the planetary gear set. Under the ideal condition where the deformations of bearings and gear teeth are not considered, the movement of the \(i\)th planet and the sun gear can be regarded as fixed-axis rotation. The angles fit a transmission ratio equation as follows:  
\[\left(\theta_{P}^{i}-\Delta\gamma_{PS}^{i}\right)\cdot Z_{P}+\left(\theta_{S}- \Delta\gamma_{PS}^{i}\right)\cdot Z_{S}=0, \tag{2}\]  
where \(\gamma_{PS}^{i}=\arctan\left[\left(y_{P}^{i}-y_{S}\right)/\left(x_{P}^{i}-x_{S} \right)\right]\) is the time-varying position angle between the center connection line \(O_{P}^{i}O_{S}\) and X axis, \(\Delta\gamma_{PS}^{i}=\gamma_{PS}^{i}-\gamma_{PS0}^{i}\), and \(Z\) denotes the tooth number. However, when considering the flexibility of the gear body, the deflection of gear teeth and the deformation of gear contact, the preceding equation does not fit anymore. The ideal angle \(\theta_{P-PS}^{i}\) of the planet gear under the condition of a rotation angle \(\theta_{S}\) and a position angle \(\Delta\gamma_{PS}^{i}\) is  
\[\theta_{P-PS}^{i}=-\left(\theta_{S}-\Delta\gamma_{PS}^{i}\right)\cdot Z_{S}/Z _{P}+\Delta\gamma_{PS}^{i}. \tag{3}\]  
As a result, the angle difference \(\Delta\theta_{P-PS}^{i}\) with respect to the \(i\)th planet gear is  
\[\Delta\theta_{P-PS}^{i}=\theta_{P}^{i}-\theta_{P-PS}^{i}. \tag{4}\]  
The linear deformation caused by this angle difference is  
\[\delta_{\theta n-PS}^{i}=r_{P-PS}^{i}\cdot\tan\Delta\theta_{P-PS}^{i}\cdot\cos \alpha_{PS}^{i}, \tag{5}\]  
where \(r_{P-PS}^{i}=d_{PS}^{i}\cdot Z_{P}/\left(Z_{S}+Z_{P}\right)\) is the pitch radius of the \(i\)th planet gear, which is time-varying during the meshing process and can be derived from the center distance \(d_{PS}^{i}=\left[\left(x_{P}^{i}-x_{S}\right)^{2}+\left(y_{P}^{i}-y_{S}\right) ^{2}\right]^{1/2}\), and \(\alpha_{PS}^{i}\) in the equation is the engagement angle: \(\alpha_{PS}^{i}=\arccos[\left(r_{P}+r_{Sb}\right)/d_{PS}^{i}]\). As shown in Figure 3b, \(\Delta d_{PS}^{i}=d_{PS}^{i}-d_{PS0}^{i}\), which denotes the center distance variation of \(d_{PS}^{i}\), is another crucial factor in calculating the composite deformation. The linear deformation caused by the variation of center distance is  
\[\delta_{dn-PS}^{i}=-\Delta d_{PS}^{i}\cdot\sin\alpha_{PS}^{i}. \tag{6}\]  
The normal composite deformation of the equivalent spring damper between the planet and sun gears is therefore given by  
\[\delta_{n-PS}^{i}=\delta_{\theta n-PS}^{i}+\delta_{dn-PS}^{i}=r_{P-PS}^{i} \cdot\tan\Delta\theta_{P-PS}^{i}\cdot\cos\alpha_{PS}^{i}-\Delta d_{PS}^{i} \cdot\sin\alpha_{PS}^{i}. \tag{7}\]The tangent composite deformation that is mainly caused by variation in center distance can be represented as  
\[\delta^{i}_{t-PS}=-\Delta d^{i}_{PS}\cdot\cos\alpha^{i}_{PS}. \tag{8}\]  
Furthermore, during the gear meshing process, the stiffness of an equivalent spring damper varies with changes in the gear meshing pair number, which toggles between 1 and 2 for a spur gear and is larger for a helical gear.