# Free Vibration Analysis of a Thermally Loaded Porous Functionally Graded Rotor-Bearing System
## 3 Finite Element Modelling of Porous FG Rotor-Bearing System
### Linear Bearings  
The bearings used in this work obey the equation of motion [18] shown in Equation (19).  
\[\left[\mathrm{C}^{b}\right]\left(\dot{q}^{b}\right)\,+\,\left[\mathrm{K}^{b} \right]\left(q^{b}\right)\,=\,\left\{Q^{b}\right\} \tag{19}\]  
where \(\left[\mathrm{C}^{b}\right]\) is the damping matrix of the bearings, \(\left[\mathrm{K}^{b}\right]\) is the stiffness matrix of the bearings, \(\left\{q^{b}\right\}\) is the nodal displacement vector and \(\left\{Q^{b}\right\}\) is the external loading vector.  
\[\left\{q^{b}\right\}\,=\,\left\{\begin{array}{c}v\\ w\end{array}\right\},\,\,\,\,\left[\mathrm{C}^{b}\right]\,\,=\,\left[\begin{array} []{cc}\mathrm{c}^{b}_{vv}&\mathrm{c}^{b}_{vw}\\ \mathrm{c}^{b}_{vv}&\mathrm{c}^{b}_{ww}\end{array}\right],\,\,\,\left[ \mathrm{K}^{b}\right]\,\,=\,\left[\begin{array}{cc}\mathrm{k}^{b}_{vv}& \mathrm{k}^{b}_{vw}\\ \mathrm{k}^{b}_{vv}&\mathrm{k}^{b}_{ww}\end{array}\right] \tag{20}\]  
The bearings used in the system are isotropic in nature \(\mathrm{k}^{b}_{vw}=\mathrm{k}^{b}_{wvv}=0\) and \(\mathrm{c}^{b}_{ww}=\,\mathrm{c}^{b}_{vv}=0\).  
**4. System Equation of Motion and Solution Procedure**  
The assembled equation of motion of the FG rotor-bearing system with and without porosity is  
\[\left[\mathrm{M}_{t}\right]\left[\dot{q}\right]\,-\,\Omega\left[\mathrm{G}_{ t}\right]\left\{\dot{q}\right\}\,+\,\left[\mathrm{K}_{t}\right]\left\{q\right\}\,=\,\left\{Q\right\} \tag{21}\]  
where \(\left[\mathrm{M}_{t}\right]\) is the global mass matrix, which is incorporated into the elemental translational and rotary mass matrices of the porous FG shaft and disc mass, while \(\left[\mathrm{G}_{t}\right]\) is the global gyroscopic matrix, which includes the gyroscopic moments of the FG shaft and disc. \(\left[\mathrm{K}_{t}\right]\) is the global stiffness matrix, which includes all the elemental stiffness matrices of the porous FG shaft and the stiffness matrix \(\left[\mathrm{K}^{b}\right]\) of the bearings.  
Equation (21) can be expressed as  
\[\left[\mathrm{A}\right]\dot{h}\,+\,\left[\mathrm{B}\right]h=0 \tag{22}\]  
where  
\[\left[\mathrm{A}\right]\,=\,\left[\begin{array}{cc}0&\left[\mathrm{M}_{t} \right]\\ \left[\mathrm{M}_{t}\right]&-\Omega\left[\mathrm{G}_{t}\right]\end{array} \right],\,\left[\mathrm{B}\right]=\left[\begin{array}{cc}-\left[\mathrm{M}_{ t}\right]&0\\ 0&\left[\mathrm{K}_{t}\right]\end{array}\right],\,h=\left\{\begin{array}{c} \dot{q}\\ q\end{array}\right\}\]  
Equation (23) is assumed as the solution of Equation (22)  
\[h=\mathrm{h}_{\mathrm{o}}\mathrm{e}^{\lambda t} \tag{23}\]  
Equation (24) can be obtained by substituting Equation (23) into Equation (22),  
\[\lambda\left[\mathrm{A}\right]\mathrm{h}_{\mathrm{o}}\mathrm{e}^{\lambda t}\, +\,\left[\mathrm{B}\right]\mathrm{h}_{\mathrm{o}}\mathrm{e}^{\lambda t}=0 \tag{24}\]  
Eigen values can be calculated by solving the Eigen value problem; see Equation (25).  
\[\left(\left[\mathrm{B}^{-1}\right]\left[\mathrm{A}\right]\,+\,\lambda\mathrm{I} \right]\mathrm{h}_{\mathrm{o}}=0 \tag{25}\]The Eigen values are of the form  
\[\lambda_{\rm n}(\Omega)\ =\ \xi_{\rm n}(\Omega)\ \pm\ {\rm i}\omega_{\rm n}(\Omega) \tag{26}\]  
where \(\ \xi\) is the damping constant and \(\omega\) is the natural frequency. The inverse of the imaginary part at \(\Omega\) (spin speed) = 0 gives the value of the natural frequency in rad/s.