# Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffness
## 2 Analytical Model
To estimate the critical speed analytically, the high-order differentials and stiffnesses of Equation (16) are ignored, and the estimated steady radial deformation can be obtained from Equation (16) as follows:  
\[u_{e}=\frac{\rho AR_{1}^{2}i_{0}^{\prime}\omega^{2}}{\frac{E\dot{A}}{\dot{R}_{1 }}-\rho AR_{1}i_{0}^{\prime}\omega^{2}} \tag{17}\]  
It can be found that the steady ring radial deformation in Equation (17) becomes extremely large when the speed, \(\omega\), is close to \(\omega_{c}\):  
\[\omega_{c}=\sqrt{EA/\rho AR_{1}^{2}i_{0}^{\prime}} \tag{18}\]  
When the speed, \(\omega\), is larger than \(\omega_{c}\), the steady status in Equation (17) is unstable. By applying the parameters in Table 1 in Section 3, the critical speed, \(\omega_{c}\), calculated by Equation (16) and Equation (18), is 5.4339 \(\times\) 10\({}^{4}\) and 5.4324 \(\times\) 10\({}^{4}\), respectively. Therefore, it is reasonable to roughly estimate the critical speed of Equation (16) by the simplified Equation (18). In this work, the speed is restricted lower than that of Equation (18). Equation (13) can be rewritten to be the equations of small vibrations about the steady status by substitution of \(u\to u_{e}+u\). After that, Equation (13) is normalized by the following variables, to investigate the general vibration properties of planet rings:  
\[\omega_{0}=\frac{1}{R_{1}^{2}}\sqrt{\frac{EI}{\rho A}},\overline{u},\overline {v}=\frac{u,v}{R_{1}},\overline{t}=\omega_{0}t,\overline{\omega},\overline{ \Omega}=\frac{\omega,\Omega}{\omega_{0}},\overline{k}_{bi},\overline{k}_{mi}= \frac{k_{bi},k_{mi}\dot{R}_{1}^{3}}{EI},\gamma=\frac{EAR_{1}^{2}}{EI} \tag{19}\]  
The dimensionless equations are obtained by substitution of Equation(19):  
\[ \begin{array}{l}\frac{\partial^{2}u}{\partial t^{2}}-2(\Omega+\omega)\frac {\partial v}{\partial t}-(\Omega+\omega)^{2}u+\gamma\left(u+\frac{\partial v}{ \partial\theta}\right)+\left(\frac{\partial^{4}u}{\partial\theta^{4}}-\frac{ \partial^{2}v}{\partial\theta^{3}}\right)\\ +\sum\limits_{i=1}^{N_{u}}[k_{pi}(\theta_{bi},t)]\Delta(\theta-\theta_{bi})+ \sum\limits_{i=1}^{N_{u}}[k_{mi}u(\theta_{mi},t)\sin{a}]\Delta(\theta-\theta_{ mi})=\frac{R_{0}}{R_{1}}\Omega^{2}\cos{\phi}\\ \\ \frac{\partial^{2}v}{\partial t^{2}}+2(\Omega+\omega)\frac{\partial u}{ \partial t}-(\Omega+\omega)^{2}v-\gamma\left(\frac{\partial u}{\partial\theta} +\frac{\partial^{2}v}{\partial\theta^{2}}\right)+\left(\frac{\partial^{2}v}{ \partial\theta^{3}}-\frac{\partial^{2}v}{\partial\theta^{2}}\right)\\ +\sum\limits_{i=1}^{N_{u}}[k_{mi}v(\theta_{mi},t)\cos{a}]\Delta(\theta-\theta_{ mi})=-\frac{R_{0}}{R_{1}}\Omega^{2}\sin{\phi}\end{array}  \tag{20a}\]  
The right parts of Equation (20) are external excitations resulting from rotating and revolving movement. In Equation (20), the bar on variables is removed for convenience, and Equation (20) is rewritten in matrix form by partial differential operators, as follows:  
\[\mathbf{M}\ddot{\mathbf{x}}+(\Omega+\omega)\mathbf{G}\dot{\mathbf{x}}+\mathbf{ K}\mathbf{x}=\mathbf{F} \tag{21}\] \begin{table}
\begin{tabular}{c c} \hline Ring radius \(R_{0}\) (mm) & 100 \\ Ring radius \(R_{1}\) (mm) & 80 \\ Cross section area \(A\) (mm\({}^{2}\)) & 250 \\ Area moment of inertia \(I\) (mm\({}^{4}\)) & 2083 \\ Pressure Angle \(\alpha\) (\({}^{\circ}\)) & 20 \\ Elastic modulus \(E\) (GPa) & 206 \\ Density \(p\) (kg/m\({}^{3}\)) & 7800 \\ Mean mesh stiffness \(k_{mi}\) (N/m) & \(5\times 10^{8}\) \\ Support stiffness \(k_{bi}\) (N/m) & \(1\times 10^{8}\) \\ The ratio \(i_{0}\) of \(\Omega\) to \(\omega\) & 0.1825 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of a planet ring with discrete and partially distributed stiffnesses. where  
\[\mathbf{x}=\left\{ \begin{array}{cc}u&v&v\end{array} \right\}^{\mathrm{T}}\!, \mathbf{M}=\left[ \begin{array}{cc}1&0\\ 0&1\end{array} \right]\!,\mathbf{G}=\left[ \begin{array}{cc}0&-2\\ 2&0\end{array} \right]\!,\mathbf{F}=\left\{\frac{R_{0}}{R_{1}}\Omega^{2}\cos \phi,-\frac{R_{0}}{R_{1}}\Omega^{2}\sin\phi\right\}^{\mathrm{T}}\!,\]  
\[\mathbf{K}=\left[