# Free Vibration Analysis of a Thermally Loaded Porous Functionally Graded Rotor-Bearing System
## 2 Materials and Methods  
Material modelling of a porous FG shaft is reported in this section. Detailed descriptions of the non-linear temperature distribution (NLTD), as well as the power law gradation of nonporous and porous FG shafts, are presented in Sections 2.1-2.3, respectively.  
Material gradation is divided into three categories: continuously graded microstructure, discretely graded microstructure and multi-phase graded microstructure [39]. Generally, in FGM, material properties (thermal conductivity K, Young's modulus E, density and Poisson's ratio) are diversified along the gradation direction. The radial direction is considered the gradation direction in the case of circular FG shafts; as the precise details of the graded microstructure of FGMare obscured, the volume fraction of different phases is used for material gradation. The distribution of metal and ceramic phases along the radial direction of the FG shaft is represented in Figure 1. The properties of FGM must be position and temperature dependent for a precise model of the FG shaft. The position dependency of material properties can be achieved by the Voigt model [40], which is a simple rule for mixtures of composites. The material properties of a specific layer \(\mathbf{P_{I}}\) are expressed as in Equation (1).  
\[P_{I}=P_{m}V_{m}+P_{c}V_{c} \tag{1}\]  
where \(P_{m}\) and \(P_{c}\) are the material properties of metal and ceramic; \(V_{m}\) and \(V_{c}\) are the volume fractions of metal and ceramic of the respective layers. The temperature dependency of material properties given by Touloukian [41] can be expressed as a non-linear function of temperature, as shown in Equation (2). Although there are many other non-linear variations in the rotating systems [42; 43], the non-linear gradation of material properties and non-linear temperature distribution are considered in the present work.  
\[P(T)\ =\ P_{0}\Big{(}P_{-1}T^{-1}+1+\ P_{1}T+\ P_{2}T^{2}+\ P_{3}T^{3}\Big{)} \tag{2}\]  
where \(T\) is temperature in Kelvin, and \(P_{-1}\), \(P_{0}\), \(P_{1}\), \(P_{2}\) and \(P_{3}\) are coefficients of temperature that are different for different material constituents. [PERSON] and [PERSON] [44] listed these coefficients and they are represented in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Material Properties**} & \multicolumn{1}{c}{\(\mathbf{P_{0}}\)} & \multicolumn{1}{c}{\(\mathbf{P_{-1}}\)} & \multicolumn{1}{c}{\(\mathbf{P_{1}}\)} & \multicolumn{1}{c}{\(\mathbf{P_{2}}\)} & \multicolumn{1}{c}{\(\mathbf{P_{3}}\)} \\ \hline \multirow{3}{*}{Steel} & E & \(201.04\times 10^{9}\) & 0 & \(3.08\times 10^{-4}\) & \(-6.534\times 10^{-7}\) & 0 \\  & K & 15.379 & 0 & \(-1.26\times 10^{-3}\) & \(2.09\times 10^{-6}\) & \(-3.7\times 10^{-10}\) \\  & v & 0.326 & 0 & \(-2\times 10^{-4}\) & \(3.797\times 10^{-7}\) & 0 \\ \hline \multirow{3}{*}{ZrO\({}_{2}\)} & E & \(244.27\times 10^{9}\) & 0 & \(-1.37\times 10^{-3}\) & \(1.21\times 10^{-6}\) & \(-3.7\times 10^{-10}\) \\  & K & 1.7 & 0 & \(1.276\times 10^{-4}\) & \(6.648\times 10^{-8}\) & 0 \\  & v & 0.2882 & 0 & \(1.133\times 10^{-4}\) & 0 & 0 \\ \hline \multirow{3}{*}{Si\({}_{3}\)N\({}_{4}\)} & E & \(348.4\times 10^{9}\) & 0 & \(-3.07\times 10^{-4}\) & \(2.160\times 10^{-7}\) & \(-8.9\times 10^{-11}\) \\  & K & & 0 & \(9.095\times 10^{-4}\) & 0 & 0 \\  & v & 0.24 & 0 & 0 & 0 & 0 \\ \hline \multirow{3}{*}{SUS304} & E & \(201.04\times 10^{9}\) & 0 & \(3.07\times 10^{-3}\) & \(-6.534\times 10^{-7}\) & 0 \\  & K & & 0 & \(1.276\times 10^{-4}\) & 0 & 0 \\ \cline{1-1}  & v & 0.3262 & 0 & \(-2\times 10^{-4}\) & \(3.797\times 10^{-7}\) & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Thermal coefficients of the constituent materials of functionally graded materials (FGMs).  
Figure 1: Distribution of metal and ceramic phases along the radial direction of a functionally graded (FG) shaft.