# 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  
The FG shaft was divided into ten finite elements. A two-node FG shaft element was developed based on Timoshenko beam theory. Each node of the element was considered to have four degrees of freedom (two translational and two rotational) as shown in Figure 3b. As the analysis was performed for natural bending frequencies, which are appropriate in the case of a Jeffcott rotor supported on linear bearings in transverse directions, the axial translational degree of freedom was not considered. The equation of motion of the uniform shaft element is mentioned in the work of [PERSON] and [PERSON] [PERSON] and [PERSON] (1966). The equation of motion of the porous FG shaft element (Equation (10)) can be developed by the application of [PERSON]'s extended principle and by using the equation of energy and work functions discussed in ([PERSON], 1969). However, in the present work, the individual matrices are also the functions of the volume fraction of porosity, the radial variation in the material properties and the temperature gradients, unlike the uniform shaft element. \[([\mathrm{M}^{\mathrm{e}}]\;+\;[\mathrm{N}^{\mathrm{e}}])[\breve{q}]\;-\; \Omega[\mathrm{G}^{\mathrm{e}}][\breve{q}]\;+\;[\mathrm{K}^{\mathrm{e}}][q]\;=\; [\mathrm{Q}_{\mathrm{c}}^{\mathrm{e}}]\mathrm{cos}\;\Omega\mathrm{t}\;+\;[ \mathrm{Q}_{\mathrm{s}}^{\mathrm{e}}]\mathrm{sin}\;\Omega\mathrm{t} \tag{10}\]  
The coefficient matrices are  
\[[\mathrm{K}^{\mathrm{e}}]\;=\;[\mathrm{K}]_{0}\;+\;\mathcal{O}_{R}[\mathrm{K} ]_{1} \tag{11a}\] \[[\mathrm{M}^{\mathrm{e}}]\;=\;[\mathrm{M}]_{0}+\;\mathcal{O}_{R}[\mathrm{M} ]_{1}+\;\mathcal{O}_{R}{}^{2}[\mathrm{M}]_{2}\] (11b) \[[\mathrm{N}^{\mathrm{e}}]\;=\;[\mathrm{N}]_{0}+\;\mathcal{O}_{R}[\mathrm{N} ]_{1}+\;\mathcal{O}_{R}{}^{2}[\mathrm{N}]_{2}\] (11c) \[[\mathrm{G}^{\mathrm{e}}]\;=\;[\mathrm{G}]_{0}+\;\mathcal{O}_{R}[\mathrm{G} ]_{1}+\;\mathcal{O}_{R}{}^{2}[\mathrm{G}]_{2} \tag{11d}\]  
Equation (10) represents the elemental equation of motion of a porous FG shaft. \(\Omega\) is the spin speed of the shaft in rad / sec. \([\mathrm{K}^{\mathrm{e}}]\) is the elemental stiffness matrix, \([\mathrm{M}^{\mathrm{e}}]\) is the elemental translational mass matrix, \([\mathrm{N}^{\mathrm{e}}]\) is the elemental rotatory matrix and \([\mathrm{G}^{\mathrm{e}}]\) is the elemental gyroscopic matrix of the porous FG shaft. \([\mathrm{K}^{\mathrm{e}}]\), \([\mathrm{M}^{\mathrm{e}}]\), \([\mathrm{N}^{\mathrm{e}}]\) are symmetric matrices, whereas \([\mathrm{G}^{\mathrm{e}}]\) is a skew symmetric matrix. \(\mathcal{O}_{R}\) is the transverse shear effect and is shown in Equation (12a). As the shaft element is composed of FGM, it also depends on radial distance, as well as the volume fraction of porosity and temperature. \[\mathcal{O}(r,T,\alpha)\;=\frac{12E(r,\;T,\alpha)\;I(r)\;dr}{\kappa(r,\;T, \alpha)\;A\;G(r,\;T,\alpha)l_{e}^{2}}\] (12a) where, \[E(r,\;T,\alpha)\] is Young's Modulus, \[I(r)\] is the moment of inertia, \[\kappa(r,\;T,\alpha)\] is the transverse shear form factor, \[G(r,\;T,\alpha)\] is the Shear Modulus of the shaft element, \[[\mathrm{K}^{\mathrm{e}}]\;=\;[\mathrm{K}]_{0}\;+\;\mathcal{O}_{R}[\mathrm{K} ]_{1} \tag{11a}\]0 is the length of the element and \[[\mathrm{K}^{\mathrm{e}}]\;=\;[\mathrm{K}]_{0}\;+\;\mathcal{O}_{R}[\mathrm{K} ]_{1} \tag{11a}\]1  
\[[\mathrm{K}^{\mathrm{e}}]\;=\;[\mathrm{K}]_{0}\;+\;\mathcal{O}_{R}[\mathrm{K} ]_{1} \tag{11a}\]2 (a) An FG rotor-bearing system. (b) A porous FG shaft element. is the thickness of the shaft element whose inner radius is \(r\) and outer radius is \(r+dr\). A solid FG shaft whose inner radius \(R_{i}=0\) and outer radius \(R_{o}=R\) is shown in Equation (12b). \[[\mathrm{K}^{\mathrm{e}}]\;=\;[\mathrm{K}]_{0}\;+\;\mathcal{O}_{R}[\mathrm{K} ]_{1} \tag{11a}\]3  
The spatial constraint matrix associated with translational shape functions is shown in Equation (13). \[[\uppsi(\mathrm{s})]\ =\ \left[