# Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffness
## 2 Analytical Model
 \tag{1}\]  
where \(\phi=\omega t+\theta,\,{\bf r}_{M}\) is the vector of \(M\) in the coordinate {e\({}_{1}\), e\({}_{2}\), e\({}_{3}\)}, and \({\bf v}_{M,r}\) is the relative velocity of \(M\) in the coordinate {e\({}_{1}\), e\({}_{2}\), e\({}_{3}\)}, which can be expressed as follows:  
\[{\bf v}_{M,r}={\bf v}_{O_{2}}+\omega{\bf e}_{3}^{\prime}\times{\bf r}_{M1}+{ \bf v}_{r}=R_{1}\omega{\bf e}_{2}^{\prime}+\omega{\bf e}_{3}^{\prime}\times \left(u{\bf e}_{1}^{\prime}+v{\bf e}_{2}^{\prime}\right)+\left(i{\bf e}_{1}^{ \prime}+i{\bf e}_{2}^{\prime}\right) \tag{2}\]  
Consequently, \({\bf v}_{M}\) becomes the following:  
\[{\bf v}_{M}=R_{0}\Omega{\bf e_{2}}+\Omega{\bf e_{3}}\times\left(R_{1}\cos\phi{ \bf e_{1}}+R_{1}\sin\phi{\bf e_{2}}+u{\bf e}_{1}^{\prime}+v{\bf e}^{\prime} \right)+\left(i-\omega v\right){\bf e}_{1}^{\prime}+\left(R_{1}\omega+\omega u +\hat{v}\right){\bf e}_{2}^{\prime} \tag{3}\]  
Since the velocity, \({\bf v}_{M}\), in Equation (3) is represented by unit vectors of two coordinates, it is better to be represented in one coordinate. Thus, according to the coordinate transformation, the velocity of \(M\) can be represented in the coordinate {e\({}_{1}\)\({}^{\prime}\), e\({}_{2}\)\({}^{\prime}\), e\({}_{3}\)\({}^{\prime}\)} as follows:  
\[ \begin{array}{ll}{\bf v}_{M}=v_{r}{\bf e}_{1}^{\prime}+v_{t}{\bf e}_{2}^{ \prime}\\ =\left[\dot{u}-v(\Omega+\omega)+R_{0}\Omega\sin\phi\right]{\bf e}_{1}^{\prime}+ \left[\dot{v}+u(\Omega+\omega)+R_{1}(\Omega+\omega)+R_{0}\Omega\cos\phi\right] {\bf e}_{2}^{\prime}\end{array}  \tag{4}\]where \(v_{r}\) is the radial velocity and \(v_{t}\) is the tangential velocity. Then the kinetic energy of the ring is as follows:  
\[K_{E}=\frac{1}{2}\rho AR_{1}\int_{0}^{2\pi}\left(v_{r}^{2}+v_{t}^{2}\right)d\theta \tag{5}\]  
Because the strain energy of the ring is independent of the rotating and revolving movement, the deformation of point \(M\) can be considered in the coordinates \(\left\{\mathbf{e}_{1}\mathbf{{}^{\prime}},\ \mathbf{e}_{2}\mathbf{{}^{\prime}},\ \mathbf{e}_{3}\mathbf{{}^{ \prime}}\right\}\). The original length of a small segment of the middle surface of the ring is given as \(\mathrm{d}S=R_{1}\mathrm{d}\theta\). After deformation, the vector \(O_{1}M\), which is denoted by \(\mathbf{r}\) and \(\mathbf{r}=(R_{1}+u)\mathbf{e}_{1}\mathbf{{}^{\prime}}+\mathbf{e}_{2}\mathbf{{ }^{\prime}}\), will increase by \(\mathrm{d}\mathbf{r}\). Then the length of the segment becomes the following:  
\[\mathrm{d}S^{\prime}=\sqrt{\mathrm{d}\mathbf{r}\cdot\mathrm{d}\mathbf{r}}= \sqrt{\left[(R_{1}+u+\partial v/\partial\theta)\mathrm{d}\theta\right]^{2}+ \left[(\partial u/\partial\theta-v)\mathrm{d}\theta\right]^{2}} \tag{6}\]  
Therefore, the strain of the segment on the middle surface is as follows:  
\[\varepsilon_{0}=\frac{\mathrm{d}S^{\prime}-\mathrm{d}S}{\mathrm{d}S}=\frac{ \sqrt{\left[(R_{1}+u+\partial v/\partial\theta)\mathrm{d}\theta\right]^{2}+ \left[(\partial u/\partial\theta-v)\mathrm{d}\theta\right]^{2}}-R_{1} \mathrm{d}\theta}{R_{1}\mathrm{d}\theta} \tag{7}\]  
Here, Equation (7) is approximated to second-order, and it gives the following:  
\[\varepsilon_{0}=\frac{1}{R_{1}}\left(u+\frac{\partial v}{\partial\theta} \right)+\frac{1}{2R_{1}^{2}}\left(v-\frac{\partial u}{\partial\theta}\right)^ {2} \tag{8}\]  
The strain of any material point on the cross-section of the ring can be obtained by linear assumption:  
\[\varepsilon=\varepsilon_{0}+\frac{r^{\prime}}{R_{1}^{2}}\left(\frac{\partial v }{\partial\theta}-\frac{\partial^{2}u}{\partial\theta^{2}}\right) \tag{9}\]  
where \(r^{\prime}\) is the radial distance of a point on the cross-section to the middle surface. Therefore, the total strain energy of the ring is as follows:  
\[S_{e}=\frac{1}{2}\iint\limits_{A\theta}\sigma edAR_{1}\mathrm{d}\theta \tag{10}\]  
where \(\sigma\) is the stress.