# Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffness
## 2 Analytical Model
The actions of partially distributed and discrete stiffnesses are considered as external forces to the ring; therefore, the work of the stiffnesses is as follows:  
Figure 1: Schematic of planetary gear set: (**a**) physical model, 1—sun gear, 2—planet gear, 3—carrier, 4—ring gear, 5—bearing, 6—planet gear shaft; (**b**) vibration model of a planet ring. \[W_{s}=\int_{\theta}[(F_{b}+F_{mr})u+F_{mt}v]d\theta \tag{11}\]  
where \(F_{b}\) is elastic forces of partially distributed stiffnesses, and \(F_{mr}\) and \(F_{mt}\) are radial and tangential components of discrete stiffnesses acting on the ring, respectively. Accordingly, the governing equations can be derived by [PERSON]'s principle:  
\[\delta\int_{t_{0}}^{t_{1}}(K_{E}-S_{e}+W_{s})\mathrm{d}t=0 \tag{12}\]  
By substituting Equations (5), (10), and (11) into Equation (12) and ignoring the nonlinear terms for the small vibrations, we get the governing equations of vibration of the planet ring:  
\[ \begin{array}{l}\rho AR_{1}\frac{\partial^{2}u}{\partial t^{2}}-2\rho AR_{1} (\Omega+\omega)\frac{\partial v}{\partial t}-\rho AR_{1}(\Omega+\omega)^{2}u +N-\frac{\partial^{2}M}{R_{1}\partial\theta^{2}}+F_{b}+F_{mr}\\ =\rho AR_{1}^{2}(\Omega+\omega)^{2}+\rho AR_{1}R_{0}\Omega^{2}\cos\phi\end{array}  \tag{13a}\] \[ \begin{array}{l}\rho AR_{1}\frac{\partial^{2}v}{\partial t^{2}}+2\rho AR_{1} (\Omega+\omega)\frac{\partial u}{\partial t}-\rho AR_{1}(\Omega+\omega)^{2}v -\frac{\partial N}{\partial t}-\frac{\partial M}{R_{1}\partial\theta}+F_{mt} \\ =-\rho AR_{1}R_{0}\Omega^{2}\sin\phi\end{array}  \tag{13b}\]  
where \(N\) and \(M\) are the vibration-induced force and moment resultants, respectively:  
\[N=EA\bigg{(}\frac{u}{R_{1}}+\frac{\partial v}{R_{1}\partial\theta}\bigg{)} \tag{14a}\] \[M=\frac{EI}{R_{1}^{2}}\bigg{(}\frac{\partial v}{\partial\theta}-\frac{\partial^ {2}u}{\partial\theta^{2}}\bigg{)} \tag{14b}\]  
and  
\[F_{b}=\sum_{i=1}^{N_{b}}[F_{bi}\Delta(\theta-\theta_{bi})]=\sum_{i=1}^{N_{b}} [k_{bi}u(\theta_{bi},t)]\Delta(\theta-\theta_{bi}) \tag{15a}\] \[F_{mr}=\sum_{i=1}^{N_{m}}[F_{mri}\Delta(\theta-\theta_{mi})]=\sum_{i=1}^{N_{m}} [k_{mi}u(\theta_{mi},t)\sin\alpha]\Delta(\theta-\theta_{mi})\] (15b) \[F_{mt}=\sum_{i=1}^{N_{m}}[F_{mri}\Delta(\theta-\theta_{mi})]=\sum_{i=1}^{N_{m}} [k_{mi}v(\theta_{mi},t)\cos\alpha]\Delta(\theta-\theta_{mi}) \tag{15c}\]  
\(F_{bi}\), \(F_{mri}\), and \(F_{mri}\) are forces of the \(i\)th partially distributed stiffness, radial, and tangential components of discrete stiffness, respectively. \(N_{b}\) and \(N_{m}\) are the numbers of partially distributed stiffnesses and discrete stiffnesses, respectively. \(\theta_{bi}\), \(\theta_{mi}\) are the corresponding time dependent positions of partially distributed stiffnesses and discrete stiffnesses, respectively. \(\Delta\) (\(\cdot\)) is the Dirac delta function. It can be seen from the right side of Equation (13) that the rotation and revolving movements will result in constant centrifugal force, \(\rho AR_{1}^{2}(\Omega+\omega)^{2}\), and time-dependent excitation, \(\rho AR_{1}R_{0}\Omega^{2}\cos\phi\), \(\rho AR_{1}R_{0}\Omega^{2}\sin\phi\), to the ring. For the steady configuration, the ring has no tangential deflection because there is no steady force in this direction, as shown in Equation (13b). Therefore, the radial steady equations of the ring can be obtained from Equation (13a) by the elimination of the time derivatives:  
\[ \begin{array}{l}\frac{EI}{R_{1}^{2}}\frac{\partial^{4}u_{c}}{\partial\theta ^{4}}+\Big{(}\frac{EA}{R_{1}}-\rho AR_{1}i_{0}^{\prime}\omega^{2}\Big{)}u_{c }+\sum_{i=1}^{N_{b}}[k_{bi}u_{e}(\theta_{bi})]\Delta(\theta-\theta_{bi})\\ +\sum_{i=1}^{N_{m}}[k_{mi}u_{e}(\theta_{mi})\sin\alpha]\Delta(\theta-\theta_{ mi})=\rho AR_{1}^{2}i_{0}^{\prime}\omega^{2}\end{array}  \tag{16}\]where \(u_{e}\) denotes the steady radial deformation and \(i_{0}^{\prime}=\left(i_{0}+1\right)^{2}\). Moreover, \(u_{e}\) in the nonlinear Equation (16) can be solved by numerical method.