# 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
According to the parameter identification of gear meshing stiffness, the relationship between the single pair and double pair gear meshing stiffness can be represented by a coefficient \(\xi\) (generally \(1<\xi<2\)). Starting from the moment that a single meshing begins, the time-varying meshing stiffness in one cycle is  
\[k_{PS}= \begin{cases}k&0\leq\theta\leq(2-\varepsilon)\;\theta_{T}\\ gk&(2-\varepsilon)\;\theta_{T}<\theta\leq\theta_{T}\end{cases} , \tag{9}\]  
where \(\theta\) is the meshing phase angle, and \(\theta_{T}\) is the cycle phase angle. In addition, exchanges between the single and double pair of different planet-sun gear pairs are asynchronous during the process of loading sharing. Thus, the initial phase angles \(\varphi_{i}\) representing the meshing state are introduced. Supposing that the total number of planets is \(n\), and under the premise that design, manufacturing and assembly errors are ignored, the initial phase angle of an \(i\)th planet gear that is different from the adjacent one is  
\[\varphi_{i}=(n-1)\varphi_{50}\;\&\;\varphi_{50}=\left[\frac{Z_{S}}{n}-\text{ int}\left(\frac{Z_{S}}{n}\right)\right]\cdot\frac{2\pi}{Z_{S}}. \tag{10}\]  
Thus, the time-varying meshing stiffness of the \(i\)th planet-sun gear meshing pair is  
\[k^{i}_{PS}= \begin{cases}k&(i-1)\varphi_{50}\leq\theta\leq(i-1)\varphi_{50}+(2- \varepsilon)\;\;\theta_{T}\\ \xi k&0<\theta<(i-1)\varphi_{50}\quad\text{or}\quad(i-1)\varphi_{50}+(2- \varepsilon)\;\;\theta_{T}<\theta\leq\theta_{T}\end{cases} . \tag{11}\]  
Based on the above derivations of the time-varying stiffness and composite deformations in the normal and tangent directions, the normal force exerted on the \(i\)th planet gear by the sun gear in Figure 2 is calculated as  
\[ \begin{array}{l}N^{i}_{SP}=k^{i}_{PS}\cdot\delta^{i}_{n-PS}\\ =k^{i}_{PS}\cdot\left[\frac{Z_{P}(\varphi_{P}+\varphi_{50})}{Z_{S}+Z_{P}}\tan( \theta^{i}_{P}+\frac{Z_{S}}{Z_{P}}\theta^{i}_{S}-\frac{Z_{S}+Z_{P}}{Z_{P}} \Delta\gamma^{i}_{PS})-(d^{i}_{PS}-d^{i}_{PS0})\cdot\sin\alpha^{i}_{PS}\right] \end{array} . \tag{12}\]  
The direction of this force is tangent to the base circle of the planet gear. The angle between it and the X axis is  
\[\beta^{i}_{SP-N}=\pi/2+\gamma^{i}_{SP}-\alpha^{i}_{PS}. \tag{13}\]  
The tangent friction force between the planet and sun gear is composed of the viscous damping and coulomb friction forces:  
\[f^{i}_{SP}=(\mu_{PS}N^{i}_{SP}+c_{PS}\cdot\dot{\delta}^{i}_{t-PS})\cdot\text{ sgn}(\dot{\delta}^{i}_{t-PS}). \tag{14}\]  
In this equation, \(\mu_{PS}\) is the coulomb friction coefficient, \(c_{PS}\) is the viscous damping coefficient, and \(\text{sgn}\left(x\right)\) is the sign function. The tangent friction force is perpendicular to the normal force, direction of which is opposite to the direction of the relative sliding velocity. The angle between it and the X axis is  
\[\beta^{i}_{SP-f}=\gamma^{i}_{SP}-\alpha^{i}_{PS}. \tag{15}\]