# Free Vibration Analysis of a Thermally Loaded Porous Functionally Graded Rotor-Bearing System
## 2 Materials and Methods
### Material Gradation of Porous FG Shaft  
The porosity of an FG system can be classified into even porosity and uneven porosity. The even distribution of the porosities is represented in Figure 2. The distribution of porosity in an FG rectangular beam is available in the literature [36]; however, in the present work, porosities are distributed in FG circular shafts. An FG shaft whose inner core is comprised of stainless steel (SS) and an outer layer made of ceramic (ZrO\({}_{2}\)) is considered to have porosities that are evenly distributed within the cross-section of the shaft. Typically, these porosities are formed in the shaft during the process of fabrication. Due to these porosities, the material properties of layer \(P_{l}\) are modified as shown in Equation (6).  
\[P_{l}=P_{m}\bigg{(}V_{m}-\frac{\alpha}{2}\bigg{)}+P_{c}\bigg{(}V_{c}-\frac{ \alpha}{2}\bigg{)} \tag{6}\]  
where \(\alpha\) (\(\ll 1\)) is the porosity volume fraction for the porous FG shafts and \(\alpha=0\) for the FG shaft without porosity. Here, in Equation (6), \(P_{m}\), \(P_{c}\) and \(V_{m}\), \(V_{c}\) represent material properties and volume fraction of metal and ceramic. The sum of the volume fractions of metal and ceramic is shown in Equation (7).  
\[V_{m}+V_{c}=1 \tag{7}\]Since the FG shaft consists of porosities, the material properties of the FG shaft also depend on the volume fraction of the porosity. Hence, the modified power law for even porosity is expressed as shown in Equation (8).  
\[P(r,T,\alpha)\ =\ P_{m}+\ (P_{c}-P_{m})\bigg{[}\frac{r-R_{i}}{R_{o}-R_{i}}\bigg{]}^{k }-\ (P_{c}+P_{m})\frac{\alpha}{2} \tag{8}\]  
The material properties such as [PERSON]'s modulus, [PERSON]'s ratio and the material density of imperfect FGM with an even porosity are expressed as shown in Equations (9a)-(9c).  
\[E(r,T,\alpha)\ =\ E_{m}(T)\ +\ (E_{c}(T)\ -E_{m}(T))\bigg{[}\frac{r-R_{i}}{R_{o}-R _{i}}\bigg{]}^{k}-\ (E_{c}(T)+E_{m}(T))\frac{\alpha}{2} \tag{9a}\] \[\rho(r,T,\alpha)\ =\rho_{m}(T)\ +\ (\rho_{c}(T)\ -\rho_{m}(T))\bigg{[}\frac{r-R_{i}}{R _{o}-R_{i}}\bigg{]}^{k}-\ (\rho_{c}(T)+\rho_{m}(T))\frac{\alpha}{2}\] (9b) \[v(r,T,\alpha)\ =\ v_{m}(T)\ +\ (v_{c}(T)\ -v_{m}(T))\bigg{[}\frac{r-R_{i}}{R _{o}-R_{i}}\bigg{]}^{k}-\ (v_{c}(T)+v_{m}(T))\frac{\alpha}{2} \tag{9c}\]  
where \(R_{i}\) represents the inner radius, \(R_{o}\) represents the outer radius of the shaft, \(r\) is the radial coordinate of the shaft and \(k\) is the power law index.  
A porous, functionally graded rotor-bearing system is considered in the present work. The rotor shaft is made of an FGM; it is assumed that porosity is evenly distributed in the FG shaft. The disc, made of uniform steel, is precisely located at the mid-span of the FG shaft. The ends of the shaft are mounted on linear bearings. Python (IDLE Python 3.6.1, 64-bit, Python Software Foundation, Wilmington, DE, USA) code is developed to assign the material properties using the power law and NLTD. Finite element modelling of the porous FG shaft element provides the novelty of the current research work, and is described in Section 3.