# Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffness
## 2 Analytical Model
### Natural Characteristics  
To obtain natural frequencies and modes of the planet ring, the external excitation, \(\mathbf{F}\), is set to 0. By using a separable solution \(\mathbf{x}=\mathbf{x}\mathbf{e}^{\lambda t}\), the corresponding eigenvalue problem becomes the following:  
\[\lambda^{2}\mathbf{M}_{\mathbf{X}}+\lambda(\Omega+\omega)\mathbf{G}\mathbf{x} +\mathbf{K}_{\mathbf{X}}=0 \tag{22}\]  
For the non-axisymmetric discrete and partially distributed stiffnesses on the planet ring, the eigenvectors are not single nodal diameter (ND) modes anymore. Therefore, [PERSON]'s method is applied to solve the natural frequencies and their modes for this non-axisymmetric case. The eigenvectors are given as a series of linearly independent vectors:  
\[\mathbf{x}=\left\{\begin{array}{cc}\sum\limits_{n=-N}^{N}c_{n}\phi_{n}& \sum\limits_{m=-M}^{M}d_{m}\varphi_{m}\end{array}\right\}^{\mathrm{T}} \tag{23}\]  
where \(2N+1\) and \(2M+1\) are the numbers of orders of radial basis function, \(\phi_{n}\), and tangential base function, \(\varphi\varphi_{m}\), in Equation (23), respectively. The weight function \(\mathbf{\chi}^{{}^{\prime}}=\left[\begin{array}{cc}\overline{\phi}_{j}&0; &0&\overline{\varphi}_{k};\end{array}\right]^{\mathrm{T}}\) is used to construct the Galerkin target function, and the equations are expressed in matrix form, with a size of \(2\)\((N+M+1)\)\(\times\)\(2(N+M+1)\):  
\[\lambda^{2}[\mathbf{M}]\mathbf{x}+\lambda(\Omega+\omega)[\mathbf{G}]\mathbf{x }+[\mathbf{K}]\mathbf{x}=0 \tag{24}\]  
where \(\mathbf{x}=\{c_{-N},\dots,c_{N},d_{-M},\dots,d_{M}\}^{\mathrm{T}}\),  
\[[\mathbf{M}]=\left[\begin{array}{cc}[\mathbf{M}_{u}]&0\\ 0&[\mathbf{M}_{v}]\end{array}\right]\!,[\mathbf{G}]=\left[\begin{array}{cc} 0&-[\mathbf{G}_{uv}]\\ [\mathbf{G}_{vu}]&0\end{array}\right]\!,[\mathbf{K}]=\left[\begin{array}{cc} [\mathbf{K}_{u}+\mathbf{K}_{b}+\mathbf{K}_{mr}]&[\mathbf{K}_{uv}]\\ [\mathbf{K}_{vu}]&[\mathbf{K}_{v}+\mathbf{K}_{mt}]\end{array}\right]\!,\]  
\[\mathbf{M}_{u}(j,n)=\int_{0}^{2\pi}\overline{\phi}_{j}\phi_{n}\mathrm{d}\theta,\mathbf{M}_{v}(k,m)=\!\int_{0}^{2\pi}\overline{\phi}_{k}\varphi_{m}\mathrm{d}\theta,\  
\[\mathbf{K}_{b}(j,n)=\int_{0}^{2\pi}\sum\limits_{i=1}^{N_{b}}k_{mi}\overline{\phi}_{ j}(\theta_{bi},t)\phi_{n}(\theta_{bi},t)\mathrm{d}\theta,\]  
\[\mathbf{K}_{mr}(j,n)=\int_{0}^{2\pi}\sum\limits_{i=1}^{N_{b}}k_{mi}\sin\alpha \overline{\phi}_{j}(\theta_{mi},t)\phi_{n}(\theta_{mi},t)\mathrm{d}\theta,\]  
\[\mathbf{K}_{mt}(k,m)=\int_{0}^{2\pi}\sum\limits_{i=1}^{N_{m}}k_{mi}\cos\alpha \overline{\phi}_{k}(\theta_{mi},t)\phi_{m}(\theta_{mi},t)\mathrm{d}\theta\]  
After substituting Equation (23) into Equation (22), the order numbers of radial and circumferential basis functions are \(2N+1\) and \(2M+1\), respectively. According to the [PERSON] method, each formula is multiplied by the conjugate function of each corresponding basis function and then integrated. Thus, \(2\left(N+M+1\right)\times 2\left(N+M+1\right)\) formulas can be obtained, namely the size of matrices \(\left[\mathbf{M}\right]\), \(\left[\mathbf{G}\right]\), and \(\left[\mathbf{K}\right]\) are \(2\left(N+M+1\right)\times 2\left(N+M+1\right)\). Each element of these matrices can be calculated according to the element formula under Equation (24). Choosing proper basis functions \(\phi_{n}\left(\theta\right)\) and \(\varphi_{m}\left(\theta\right)\), the eigenvalue problem of Equation (24) can be solved following the state space eigenvalue problem, as described in Reference [30], and the numerical calculation can be made by standard methods.