# Free Vibration Analysis of a Thermally Loaded Porous Functionally Graded Rotor-Bearing System
## 1 Introduction  
An important aspect to take into consideration for a superior structural performance is the material strength of the system. Traditional composite materials are impotent when they are subjected to thermo-mechanical loading, due to inter-laminar stresses which cause the de-lamination of layers. Generally, metals are preferred because of their high strength and toughness [1]. However, at high temperatures, the strength of the metal drastically deteriorates. On the other hand, ceramics are heat resistant; however, their applications are restricted due to their low toughness. The development of a new class of composites, functionally graded materials (FCMs), mitigated the problems of de-bonding, de-lamination and residual stresses in fibre-reinforced composites at elevated temperature, while making use of the advantages of both metal and ceramic material properties. FGM is an inhomogeneous micromechanical composite typically made from different phases of metal and ceramic material constituents. The volume fraction of constituent materials is arranged in the desired direction based on material laws for smooth and continuous change from one layer to another. The term FGM was coined by a group of Japanese scientists in the mid-1980s. There are numerous types of methods based on material applications to fabricate FGMs--they include plasma spraying, chemical vapour deposition, physical vapour deposition, slurry dipping, electroplating, sintering, combustion synthesis, centrifugal casting, tape casting, hot pressing, powder metallurgy, etc. [PERSON] and [PERSON] [2] and [PERSON] et al. [3] discussed some of the manufacturing processes of functionally graded materials. The initial functionally graded materials were made for the aerospace industry, as the highest temperature of the surface of spacecrafts is estimated to reach as high as 2100 K; hence, the material used on the surface should withstand that high temperature [4]. Moreover, on account of wide range of applications in many other industries, such as automotive, electronics, biomedical, marine, defence and construction, functionally graded materials have become a topic of interest for many researchers. Most of the previous works have reported on the dynamic characteristics of beams. Since the modelling of the FG shaft in the present work is based on Timoshenko beam theory with rotating effects, the literature review is arranged around the dynamic analysis of FG beams, FG rotors and then porous FG beams and rotors. Many investigations were executed to analyse FG structures, and excellent review papers are reported in the literature [5; 6; 7]. [PERSON] et al. [8] developed a shell element to analyse the nonlinear response of FG structures. [PERSON] and [PERSON] [9] performed an isogeometric analysis of FG sandwich plates based on layer wise displacement theory. [PERSON] [10] investigated a static analysis of a functionally graded simply supported beam under a uniformly distributed load by the Ritz method, using Timoshenko and higher order shear deformation beam theories. The fundamental frequency of FG beams using classical, first order and different higher order shear deformation theories for different boundary conditions was analysed [11]. [PERSON] and [PERSON] [12] performed a free vibration analysis of simply supported FG beams using parabolic shear deformation beam theory, classical beam theory and exponential shear deformation beam theory. [PERSON] and [PERSON] [13] performed a static analysis of an FG bi-directional circular beam based on Euler-Bernoulli theory. The free vibration analysis of [PERSON] and [PERSON] FG beams using the Rayleigh-Ritz method subjected to different boundary conditions is investigated by [PERSON] and [PERSON] [14]. [PERSON] et al. [15] developed a beam model for the accurate analysis of FG beams based on a material-based higher-order shear theory. Rotor-bearing systems play a crucial role in the mechanical industry, and several researchers analysed rotor-bearing systems based on various models.