# Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffness
## Appendix A Matrices and Vectors for Equation (26)  
\[\left[\widetilde{\mathbf{M}}\right]=\left[\begin{array}{ccc}1&0&0\\ 0&\ddots&0\\ 0&0&1\end{array}\right]_{4N_{f}\times 4N_{f}},\left[\overline{\mathbf{G}} \right]=\left[\begin{array}{ccc}0&0&\overline{\mathbf{G}}_{p\xi}&0\\ 0&0&\overline{\mathbf{G}}_{qq}\\ -\overline{\mathbf{G}}_{p\xi}&0&0&0\\ 0&-\overline{\mathbf{G}}_{qq}&0&0\end{array}\right]_{4N_{f}\times 4N_{f}}, \widetilde{\mathbf{K}}=\left[\begin{array}{ccc}\overline{\mathbf{K}}_{pp}& \overline{\mathbf{K}}_{pq}&0&\overline{\mathbf{K}}_{pq\eta}\\ \overline{\mathbf{K}}_{qq}&\overline{\mathbf{K}}_{q\xi}&0\\ sym&&\overline{\mathbf{K}}_{q\xi}&0\\ \overline{\mathbf{K}}_{q\eta}&\overline{\mathbf{K}}_{q\eta}\end{array}\right]_ {4N_{f}\times 4N_{f}},\]  
\[\overline{\mathbf{G}}_{p\xi}(n,k)=\left\{\begin{array}{l}-2(\Omega+\omega), n=k\\ 0,n\
eq k\end{array}\right.,\overline{\mathbf{G}}_{q\eta}=\overline{\mathbf{G}}_{p \xi},\]  
\[\overline{\mathbf{K}}_{pp}(n,k)=\left\{\begin{array}{l}\gamma-(\Omega+\omega )^{2}+n^{4}+N_{c}^{\prime}n^{2}+k_{b}\sum\limits_{i=1}^{N_{b}}\cos^{2}n\theta _{bi},n=k\\ k_{b}\sum\limits_{i=1}^{N_{b}}\cos k\theta_{bi}\cos n\theta_{bi}+\sum\limits_{i=1 }^{N_{b}}k_{mi}\sin\alpha\cos k\theta_{mi}\cos n\theta_{mi},n\
eq k\end{array} \right.,\]  
\[\overline{\mathbf{K}}_{qq}(n,k)=\left\{\begin{array}{l}\gamma-(\Omega+\omega )^{2}+n^{4}+N_{c}^{\prime}n^{2}+k_{b}\sum\limits_{i=1}^{N_{b}}\sin^{2}n\theta _{bi},n=k\\ k_{b}\sum\limits_{i=1}^{N_{b}}\sin k\theta_{bi}\sin n\theta_{bi}+\sum\limits_{i=1 }^{N_{b}}k_{mi}\sin\alpha\sin k\theta_{mi}\sin n\theta_{mi},n\
eq k\end{array} \right.,\]  
\[\overline{\mathbf{K}}_{q\xi}(n,k)=\left\{\begin{array}{l}(\gamma+1)n^{2}-( \Omega+\omega)^{2}+\sum\limits_{i=1}^{N_{b}}k_{mi}\cos^{2}n\theta_{mi},n=k\\ \sum\limits_{i=1}^{N_{b}}k_{mi}\cos\alpha\cos k\theta_{mi}\cos n\theta_{mi},n \
eq k\end{array}\right.,\overline{\mathbf{K}}_{q\eta}(n,k)=\left\{\begin{array}[ ]{l}(\gamma+1)n^{2}-(\Omega+\omega)^{2}+\sum\limits_{i=1}^{N_{b}}k_{mi}\sin^{2 }n\theta_{mi},n=k\\ \sum\limits_{i=1}^{N_{b}}k_{mi}\cos\alpha\sin k\theta_{mi}\sin n\theta_{mi},n \
eq k\end{array}\right.,\]\[\overline{\mathbf{K}}_{pq}(n,k)=k_{b}\sum_{i=1}^{N_{b}}\sin k\theta_{bi}\cos n \theta_{bi}+\sum_{i=1}^{N_{b}}k_{mi}\sin\alpha\sin k\theta_{mi}\cos n\theta_{mi},\]  
\[\overline{\mathbf{K}}_{pq}(n,k)=k_{b}\sum_{i=1}^{N_{b}}\sin k\theta_{bi}\cos n \theta_{bi}+\sum_{i=1}^{N_{b}}k_{mi}\sin\alpha\sin k\theta_{mi}\cos n\theta_{ mi},\]  
\[\overline{\mathbf{K}}_{pq}(n,k)=\left\{\begin{array}{c}\gamma n+n^{3},n=k\\ 0,n\
eq k\end{array}\right.,\overline{\mathbf{K}}_{qg}(n,k)=\left\{\begin{array} []{c}-(\gamma n+n^{3}),n=k\\ 0,n\
eq k\end{array}\right.,\]  
\[\overline{\mathbf{Q}}=\left\{\begin{array}{c}\mathbf{Q}_{c}\quad\mathbf{Q }_{2}\end{array}\right\}_{4N_{f}},\mathbf{Q}_{c}=\left\{\begin{array}{c} \mathbf{Q}_{p}\quad\mathbf{Q}_{\eta}\end{array}\right\}_{2N_{f}}^{\mathrm{T }},\mathbf{Q}_{s}=\left\{\begin{array}{c}\mathbf{Q}_{q}\quad\mathbf{Q}_{ \xi}\end{array}\right\}_{2N_{f}}^{\mathrm{T}},\]  
\[\mathbf{Q}_{p}=\left\{0\begin{array}{c}\frac{R_{0}}{R_{1}}\Omega^{2}\cos \omega t\;\;0\;\;\cdots\;0\end{array}\right\}_{N_{f}}^{\mathrm{T}},\mathbf{Q }_{q}=\left\{0\begin{array}{c}-\frac{R_{0}}{R_{1}}\Omega^{2}\sin\omega t\; \;0\;\;\cdots\;0\end{array}\right\}_{N_{f}}^{\mathrm{T}},\mathbf{Q}_{\xi}=- \mathbf{Q}_{q},\mathbf{Q}_{\eta}=\mathbf{Q}_{p},\]  
where \(n,k=0,1,2,\ldots,N_{f}-1\).