## 2 Equation of Motion for Nuclear Rod Bundles Surrounded by Viscous Fluid
### Numerical Model of Nuclear Rod Bundles
\begin{array}{ccccc}\beta_{1,1}&\beta_{1,2}&\cdots& \beta_{1,4N}\\ -\mathrm{i}\mathrm{k}_{1}\beta_{1,1}&-\mathrm{i}\mathrm{k}_{2}\beta_{1,2}&- \mathrm{i}\mathrm{k}_{4N}\beta_{1,4N}\\ \vdots&\vdots&\vdots&\vdots\\ \beta_{4N,1}&\beta_{4N,2}&\beta_{4N,4N}\\ -\mathrm{i}\mathrm{k}_{1}\beta_{4N,1}&-\mathrm{i}\mathrm{k}_{2}\beta_{4N,2}& \cdots&-\mathrm{i}\mathrm{k}_{4N}\beta_{4N,4N}\\ \beta_{1,1}e^{-\mathrm{i}\mathrm{k}_{1}\beta_{1,1}L}&\beta_{1,2}e^{-\mathrm{i }\mathrm{k}_{2}\beta_{1,2}L}&\cdots&\beta_{1,4N}e^{-\mathrm{i}\mathrm{k}_{4N} \beta_{1,4N}L}\\ -\mathrm{i}\mathrm{k}_{1}\beta_{1,1}e^{-\mathrm{i}\mathrm{k}_{1}\beta_{1,1}L} &-\mathrm{i}\mathrm{k}_{2}\beta_{1,2}e^{-\mathrm{i}\mathrm{k}_{2}\beta_{1,2}L }&-\mathrm{i}\mathrm{k}_{4N}\beta_{1,4N}e^{-\mathrm{i}\mathrm{k}_{4N}\beta_{1,4N}L}\\ \vdots&\vdots&\vdots&\vdots\\ \beta_{4N,1}e^{-\mathrm{i}\mathrm{k}_{1}\beta_{4N,1}L}&\beta_{4N,2}e^{-\mathrm{ i}\mathrm{k}_{2}\beta_{4N,2}L}&\beta_{4N,4N}e^{-\mathrm{i}\mathrm{k}_{4N} \beta_{4N,4N}L}\\ -\mathrm{i}\mathrm{k}_{1}\beta_{4N,1}e^{-\mathrm{i}\mathrm{k}_{1}\beta_{4N,1} L}&-\mathrm{i}\mathrm{k}_{2}\beta_{4N,2}e^{-\mathrm{i}\mathrm{k}_{2}\beta_{4N,2}L }&\cdots&-\mathrm{i}\mathrm{k}_{4N}\beta_{4N,4N}e^{-\mathrm{i}\mathrm{k}_{4N} \beta_{4N,4N}L}\end{array} \right].\]  
Using Equations (15) and (17), the flexural vibrations of the nuclear rod bundles in terms of the nodal displacement were obtained by  
\[W_{i}(x)=\mathbf{N}_{w,i}(x,\omega)\mathbf{d}, \tag{18}\]  
where \(\mathbf{N}_{w,i}(x,\omega)=\mathbf{e}(x,\omega)\mathbf{B}_{i}\mathbf{H}^{-1} (\omega)\). In order to obtain the spectral element equations for Equation (1), the weak form was derived from the weighted-integral statement by  
\[\int_{0}^{\pm}\left[\sum_{i=1}^{N}\left\{EIW^{\prime\prime}{}_{i}\delta W^{ \prime\prime}{}_{i}-\omega^{2}\rho_{s}A_{s}W_{i}\delta W_{i}-\omega^{2}\sum_{j =1}^{4N}M_{ij}W_{j}\delta W_{j}-F_{i,2}\delta W_{i}\right\}\right]dx=\sum_{i=1 }^{4N}F_{W,i}(x)\delta W_{i}^{l}+\sum_{i=1}^{4N}F_{\Theta,i}(x)\delta W^{ \prime}{}_{i}\right]_{0}^{l}, \tag{19}\]  
where \(\delta W\) is the arbitrary variation in \(W\), and \(F_{W,i}\) and \(F_{\Theta,i}\) are the transverse shear force and bending moment, respectively, as  
\[F_{W,i}(x)=-EI\frac{\partial^{3}W_{i}(x)}{\partial x^{3}},\]  
\[F_{\Theta,i}(x)=EI\frac{\partial^{2}W_{i}(x)}{\partial x^{2}}.\]  
Substituting Equation (18) into (19), the spectral element equation was obtained, as  
\[\mathbf{S}(\omega)\mathbf{d}=\mathbf{f}(\omega)+\mathbf{f}_{2}(\omega), \tag{20}\]  
where \(\mathbf{S}\) is the symmetric spectral element matrix for the flexural vibration of the nuclear rod bundles in the fluid as  
\[\mathbf{S}(\omega)=\mathbf{H}^{-T}\!\left(\sum_{i=1}^{N}\!\left(EI\mathbf{B}_ {i}\mathbf{K}^{2}\mathbf{E}\mathbf{K}^{2}\mathbf{B}_{i}-\rho_{s}A_{s}\omega^{ 2}\mathbf{B}_{i}\mathbf{EB}_{i}-\omega^{2}\sum_{j=1}^{N}M_{ij}\mathbf{B}_{i} \mathbf{EB}_{j}\right)\right)\!\!\mathbf{H}^{-1}.\]Here, \(\mathbf{E}(\omega)=\int_{0}^{L}\mathbf{e}^{T}(x,\omega)\mathbf{e}(x,\omega)dx\) and \(\mathbf{K}\) is the wavenumber matrix as  
\[\mathbf{K}^{2}=\left[ \begin{array}{cccc}k_{1}^{2}&&&\\ &k_{2}^{2}&&\\ &&\ddots&\\ &&&k_{4N}^{2}\end{array} \right].\]  
\(\mathbf{f}\) is the spectral nodal force component for the beam element given as  
\[\mathbf{f}(\omega)=\left\{ \begin{array}{c}\mathbf{f}_{b}(0,\omega)\\ \mathbf{f}_{b}(L,\omega)\end{array} \right\},\]  
where \(\mathbf{f}_{b}(x,\omega)=\left\{ \begin{array}{c}F_{W,1}(x,\omega)\quad F_{ \Theta,1}(x,\omega)\quad\cdots\quad F_{W,N}(x,\omega)\quad F_{\Theta,N}(x, \omega)\end{array} \right\}^{T}\), and \(\mathbf{f}_{2}\) is the spectral nodal force component induced by the pressure wave, as  
\[\mathbf{f}_{2}(\omega)=\int_{0}^{L}\sum_{i=1}^{N}F_{i,2}(\omega)\mathbf{N}_{w,j}^{T}(x,\omega)dx.\]