# Free Vibration Analysis of a Thermally Loaded Porous Functionally Graded Rotor-Bearing System
## 1 Introduction
[PERSON] [16] examined viscous and hysteretic internal damping of a rotating shaft. [PERSON] and [PERSON] [17] developed a finite element model of a turbo-rotor system by including the effects of translational kinetic energy and elastic bending energy. [PERSON] and [PERSON] [18] generalised the [PERSON]'s element by including the effects of rotatory inertia, gyroscopic moments and axial load. Extending this work, [PERSON] [19] developed a finite shaft element using Timoshenko beam theory by including the effects of transverse shear deformation. The works reported above are related to steel or uniform shafts. Very few works reported on the dynamic analysis of an FG rotor-bearing system. [PERSON] and [PERSON] [20], based on Timoshenko beam theory (TBT), presented a work on the vibration and stability analysis of a functionally graded spinning shaft using three node beam elements. [PERSON] and [PERSON] [21] performed natural frequency analysis of a functionally graded rotor system using a three-dimensional element, modelled in ANSYS (ANSYS 18.0, ANSYS, Canonsburg, PA, USA). As FGMs have an extensive variety of applications, it is important to consider the effect of temperature on FG structures and rotors. Recently, [PERSON] et al. [22] investigated the temperature-dependent free vibration analysis of FG beams with general boundary conditions. The thermo-mechanical buckling and non-vibration analysis of FG beams on a nonlinear elastic foundation are examined [23]. [PERSON] [24] analysed the thermal post buckling and nonlinear vibration behaviour of functionally graded beams based on the concept of physical neutral surfaces, von Karman strain-displacement relationships and higher order shear deformation theory. [PERSON] and [PERSON] [25] analysed the thermal buckling of FG beams. Very few works are reported to investigate the effect of temperature on FG rotors. The effect of thermal gradients on the natural frequencies of functionally graded rotor-bearing systems is investigated by [PERSON] and [PERSON] [26]. The natural frequency of the FG rotor-bearing system for different temperature gradients is analysed by [PERSON] et al. [27]. [PERSON] et al. [28] also analysed the influence of temperature gradients on the whirl frequencies of a cracked rotor. Amid all of the fabrication techniques, sintering is one of the ways to fabricate an FGM, but due to huge differences in the solidification temperatures of material constituents, micro voids and porosities are formed within the inter-layers of FGM while fabricating with this technique [29]. [PERSON] et al. [30] deduced that porosities are also formed when a functionally graded material is fabricated using a multi-step sequential infiltration technique. The density and strength of an FGM are reduced due to the presence of porosities in the structure of FGM. Since the formation of porosities is inevitable while fabricating, the effect of porosity has become extremely prominent for analysing an FG system. [PERSON] [31] explored the forced vibration analysis of FG porous deep beams under dynamic loads. [PERSON] and [PERSON] [32] investigated the vibration analysis of porous FG beams subjected to various thermal loadings based on higher order shear deformation theory carried out by utilising the Navier solution method. [PERSON] and [PERSON] [33] presented a transverse vibration analysis of rotating porous functionally graded beams using the differential transform method. [PERSON] et al. [34] presented a free vibration analysis of thick, porous, functionally graded beams resting on elastic foundations using efficient shear and normal deformation beam theory. [PERSON] and [PERSON] [35] investigated the flexural vibration analysis of a porous functionally graded beam using Timoshenko beam theory. [PERSON] and [PERSON] [36] investigated the linear and nonlinear vibration analysis of elastically end-restrained FG beams with different porosities. Researchers have been analysing FG rotor systems with crack defects.