## 2 Vibration Mechanism of Planetary Gearbox  
### Time-Varying Meshing Stiffness  
Time-varying mesh stiffness is a periodic internal excitation of gear pairs, which is very important for the vibration analysis of gear systems [18]. Considering the base deflection of the gear fillet and the influence of flexibility between the root circle and the base circle [19], the gear is simplified to a variable-section cantilever beam fixed on the gear base circle (as shown in Figure 1), and the modified potential energy method is used to calculate the time-varying meshing stiffness of the planetary gearbox for two cases, i.e., the gear root circle being smaller or bigger than the base circle, respectively [20]. According to the knowledge of principle of mechanics, the radius of gear base circle \(R_{b}\) and root circle \(R_{f}\) can be calculated as follows [21]:  
\[R_{b}=\frac{mz}{2}\cos(\theta) \tag{1}\]  
\[R_{f}=\frac{mz}{2}-({h^{*}}_{a}+c^{*})m \tag{2}\]  
where \(m\), \(z\), and \(\theta\) represent module, number of teeth, and pressure angle, respectively. \({h^{*}}_{a}\) and \(c^{*}\) are addendum and tip clearance coefficients. When the gear root circle is bigger than the base circle, the equations of \(k_{b}\), \(k_{s}\), \(k_{a}\), and \(k_{h}\) are expressed as follows [21]:  
\[k_{b,normal}=\frac{1}{\int_{-a_{1}}^{a_{4}}\frac{3\left\{1+\cos a_{1}[(a_{2} -a)\sin a-\cos a]\right\}^{2}(a_{2}-a)\cos a\right\}^{3}}\text{d}\alpha \tag{3}\]  
\[k_{s,normal}=\frac{1}{\int_{-a_{1}}^{a_{4}}\frac{1.2(1+v)(a_{2}-a)\cos a\cos a ^{2}\iota_{1}}{EL[\sin a+(a_{2}-a)\cos a]}\text{d}\alpha} \tag{4}\]  
\[k_{a,normal}=\frac{\pi a_{1}}{\int_{-a_{1}}^{a_{4}}\frac{(a_{2}-a)\cos a\sin^ {2}\iota_{1}}{2 EL[\sin a+(a_{2}-a)\cos a]}\text{d}\alpha} \tag{5}\]  
\[k_{h,normal}=\frac{\pi EL}{4(1-v^{2})}. \tag{6}\]  
When the gear root circle is smaller than the base circle, the energy stored between the base circle and the root circle (the grid line portions as shown in Figure 1) needs to be increased on the original basis; then, Equations (3)-(5) are modified as follows [21]:  
\[k_{b,normal}=\frac{1}{\int_{-a_{1}}^{a_{2}}\frac{3\left\{1+\cos a_{1}[(a_{2} -a_{1})\sin a-\cos a]\right\}^{2}(a_{2}-a)\cos a\right\}^{2}\text{d}\alpha+ \int_{0}^{\tau_{b}-\tau_{f}}\frac{\left[(d+x_{1})\cos a_{1}-h\sin a_{1}\right] ^{2}}{EL_{1}}\text{d}x_{1}} \tag{7}\]  
Figure 1: The cantilever beam model of gear teeth. \[k_{s,normal}=\frac{1}{\int_{-A_{1}}^{A_{2}}\frac{1.2(1+\varphi)(a_{2}-a)\cos a \cos^{2}a_{1}}{EL[\sin a+(a_{2}-a)\cos a]}\mathrm{d}\alpha+\int_{0}^{r_{b}-r_{f} }\frac{(1.2\cos a_{1})^{2}}{GA_{1}}\mathrm{d}x_{1}} \tag{8}\]  
\[k_{a,normal}=\frac{1}{\int_{-A_{1}}^{A_{2}}\frac{(a_{2}-a)\cos a\sin^{2}a_{1}}{ EL[\sin a+(a_{2}-a)\cos a]}\mathrm{d}\alpha+\int_{0}^{r_{b}-r_{f}}\frac{(\sin a _{1})^{2}}{EA_{1}}\mathrm{d}x_{1}}. \tag{9}\]  
The equation of \(k_{f}\) can be expressed as follows [22]:  
\[R_{f}=\frac{mz}{2}-({h^{*}}_{a}+c^{*})m \tag{2}\]0  
where \(\alpha_{m}\) is the pressure angle of contact point, \(\mu_{f}\) and \(S_{f}\) are defined as shown in \[R_{f}=\frac{mz}{2}-({h^{*}}_{a}+c^{*})m \tag{2}\]1 The coefficients \(L^{*}\), \(M^{*}\), \(P^{*}\), \(Q^{*}\) can be approached by polynomial functions [23]:  
\[R_{f}=\frac{mz}{2}-({h^{*}}_{a}+c^{*})m \tag{2}\]2  
In summary, the total effective meshing stiffness of the planetary gearbox can be described as follows:  
\[k_{total}=\left\{ \begin{array}{ll}1/\Big{(}\frac{1}{k_{b}}+\frac{1}{k_{b}}+ \frac{1}{k_{b1}}+\frac{1}{k_{b1}}+\frac{1}{k_{b1}}+\frac{1}{k_{b2}}+\frac{1}{ k_{b2}}+\frac{1}{k_{b2}}+\frac{1}{k_{b2}}+\frac{1}{k_{b2}}\Big{)},\ \text{single-tooth-pair}\\ \sum\limits_{i=1}^{2}\frac{1}{k_{b1}^{2}+\frac{1}{k_{b1}}+\frac{1}{k_{b1}}+ \frac{1}{k_{b1}}+\frac{1}{k_{b1}}+\frac{1}{k_{b2}}+\frac{1}{k_{b2}}+\frac{1}{ k_{b2}}+\frac{1}{k_{b2}}+\frac{1}{k_{b2}}},\ \text{double-tooth-pair}\end{array} \right. \tag{12}\]  
In the expressions above, subscripts 1 and 2 indicate the driving and driven gears respectively, and \(I\) denotes the gear pair number in meshing.