# Free Vibration Analysis of a Thermally Loaded Porous Functionally Graded Rotor-Bearing System
## 3 Finite Element Modelling of Porous FG Rotor-Bearing System
### Finite Porous FG Shift Element
\begin{array}{cccccccc}\uppsi_{1}&0&0&\uppsi_{2}& \uppsi_{3}&0&0&\uppsi_{4}\\ 0&\uppsi_{1}&-\uppsi_{2}&0&0&\uppsi_{3}&-\uppsi_{4}&0\end{array} \right] \tag{13}\]  
\[\uppsi_{\mathrm{j}}(\mathrm{s})\ =\ \frac{\left[\upalpha_{\mathrm{j}}(\mathrm{s})\ + \ \upPhi_{\mathrm{R}}\upbeta_{\mathrm{j}}(\mathrm{s})\right]}{1+\upPhi_{R}}\]  
\[\mathrm{j}\ =\ 1,\ 2,\ 3,\ 4;\ \mu=\frac{s}{l_{e}}\]  
\[\alpha_{1}=1-3\upmu^{2}+2\upmu^{3}\ \ \upbeta_{1}=1-\upmu\]  
\[\alpha_{2}=\mathrm{l_{e}}\upmu\big{(}1-2\upmu+\upmu^{3}\big{)}\ \ \upbeta_{2}=\mathrm{l_{e}}\upmu(1-\upmu)/2\]  
\[\alpha_{3}=\upmu^{2}(3-2\upmu)\ \ \upbeta_{3}=\upmu\]  
\[\alpha_{4}=\mathrm{l_{e}}\upmu^{2}(-1+\upmu)\ \ \upbeta_{4}=\mathrm{l_{e}} \upmu(-1+\upmu)/2\]  
The spatial constraint matrix associated with rotational shape functions is shown in Equation (14). \[[\upvarphi(\mathrm{s})]\ =\ \left[ \begin{array}{c}\upvarphi_{\mathrm{v}}\\ \upvarphi_{\mathrm{w}}\end{array} \right]\ =\ \left[ \begin{array}{cccccccc}0&-\upvarphi_{1}& \upvarphi_{2}&0&0&-\upvarphi_{3}&\upvarphi_{4}&0\\ \upvarphi_{1}&0&0&\upvarphi_{2}&\upvarphi_{3}&0&0&\upvarphi_{4}\end{array} \right] \tag{14}\]  
\[\upvarphi_{\mathrm{j}}(\mathrm{s})\ =\ \frac{\left[\upgamma_{\mathrm{j}}( \mathrm{s})+\upPhi_{\mathrm{R}}\updelta_{\mathrm{j}}(\mathrm{s})\right]}{1+ \upPhi_{\mathrm{R}}}\]  
\[\upgamma_{1}=\ \upmu(6\upmu-6)/\mathrm{l_{e}}\ \ \updelta_{1}=0\]  
\[\upgamma_{2}=1-4\upmu+3\upmu^{2}\ \ \updelta_{2}=1-\upmu\]  
\[\upgamma_{3}=\upmu(-6\upmu+6)/\mathrm{l_{e}}\ \ \updelta_{3}=0\]  
\[\upgamma_{4}=\upmu(3\upmu-2)\ \ \updelta_{4}=\upmu\]  
These spatial constraint matrices are taken from the lines of [PERSON] (1975). The shape functions are established using Timoshenko beam theory. The spatial constraint matrices, which are associated with translational and rotational shape functions, are used to derive the elemental stiffness matrix, translational mass matrix, rotatory mass matrix and gyroscopic matrix of a porous FG shaft element. These elemental matrices are derived using the equations below. Elemental stiffness matrix:  
\[[\mathrm{K^{e}}]=\mathrm{C_{1}}\left(\int\limits_{0}^{l_{e}}E(r,T,\alpha)I(r) [\psi^{\prime\prime}]^{\mathrm{T}}[\psi^{\prime\prime}]ds+\int\limits_{0}^{l_{e }}\kappa(r,\ T,\alpha)G(r,\ T,\alpha)A(r)[\psi^{\prime}]^{\mathrm{T}}[\psi^{ \prime}]ds\right)\] (15a) Elemental translatory mass matrix: \[[\mathrm{M^{e}}]\ =\ \mathrm{C_{2}}\int\limits_{0}^{l_{e}}m[\psi]^{\mathrm{T}}[\psi]ds \tag{15b}\]Elemental rotatory mass matrix:  
\[[\mathrm{N}^{\mathrm{e}}]\ =\ \mathrm{C}_{3}\int\limits_{0}^{l_{c}}I_{D}[\varphi]^{T}[ \varphi]ds \tag{15c}\]  
Elemental gyroscopic matrix:  
\[[\mathrm{G}^{\mathrm{e}}]\ =\ [\mathrm{H}^{\mathrm{e}}]\ -\ [\mathrm{H}^{ \mathrm{e}}]^{\mathrm{T}}\textit{where}\ [\mathrm{H}^{\mathrm{e}}]\ =\ \mathrm{C}_{4}\int\limits_{0}^{l_{c}}I_{P}[\varphi_{w}]^{T}[ \varphi_{v}]ds \tag{15d}\]  
While solving the elemental matrices of a porous FG shaft element, coefficients arise for every elemental matrix. \(\mathrm{C}_{1}\), \(\mathrm{C}_{2}\), \(\mathrm{C}_{3}\) and \(\mathrm{C}_{4}\) are the coefficients of the elemental stiffness matrix, elemental translatory mass matrix, elemental rotatory mass matrix and elemental gyroscopic matrix, respectively, and these coefficients depend on the radial distance, temperature and volume fraction of porosity.