## 1 Introduction  
The evolution of the latest materials with enhanced structural performance has been a topic of interest for researchers and engineers in the last few decades. Laminated fibre reinforced polymers (FRPs) are substantially used in automobile, aerospace, marine, defence, and numerous other industries. However, laminated composites are prone to debonding, delamination and residual stress under high-temperature environments. Therefore, these drawbacks limit the performance of the laminated composites in thermo-elastic applications. Functionally graded material (FGM), a new class of composite, was developed in order to mitigate these limitations. The term FGM was coined by a group of Japanese scientists in 1984. Functionally graded materials (FMMs) are high performing advanced engineering materials that are microscopically inhomogeneous composites manufactured from two or more phases of constituents. FGMs are classified into metal-ceramic, ceramic-ceramic, metal-metal and ceramic-plastic. FGMs are multifunctional composites usually made of metal and ceramic constituents. Sturdy mechanical performance for the material is provided by the metallic component, whereas ceramic constituents offer corrosion resistance and thermal resistance. The volume fractions of material constituents are varied along a particular direction based on different material gradation laws. Numerous modern fabricating techniques, such as chemical vapour deposition, centrifugal casting and spark plasma sintering, are discussed by few researchers to manufacture an FGM [1]. In the first place, FGMs were only manufactured for aircraft and spacecraft industries to sustain high temperatures since the temperature gradient of a 10 mm cross-section is estimated to reach as high as 1600 K [2], however, due to high material strength and better structural performance they are implemented in other mechanical industries. Functionally graded (FG) structures were analysed by few researchers, excellent review papers were reported in the literature [3,4]. [PERSON] [5] performed the static analysis of an FG simply supported beam using the Ritz method based on higher-order shear deformation theory and Timoshenko beam theory. [PERSON] and [PERSON] [6] analysed the natural frequencies of the FG beam based on exponential and parabolic shear deformation beam theories. [PERSON] et al. [7], based on the first-order shear deformation theory, investigated the static and free vibration of an axially loaded FG beam. [PERSON] and [PERSON] [8] developed a computational model to investigate the free vibration of an FG Timoshenko and Euler beam for various boundary conditions using the Rayleigh-Ritz method. Fundamental frequencies of an FG beam were studied by [PERSON] [9] based on classical and different higher-order shear deformation beam theories. [PERSON] [10] analysed the fundamental frequencies of a hinged and clamped FG beam by employing the finite element method and modal analysis. Rotor-bearing systems play a significant role in mining, marine, aerospace and automotive industries as the rotating shafts are one of the vital mechanical components in rotating machinery such as large turbine generators. Therefore, various researchers had investigated the vibration behaviour of the rotor system based on distinct models. [PERSON] and [PERSON] [11] generalised [PERSON]'s element by including the effects of gyroscopic moments and rotatory inertia. By including the transverse shear effects, [PERSON] [12] modelled a finite rotor element based on the Timoshenko beam theory. Few works were detailed on functionally graded rotor-bearing systems in the literature. [PERSON] and [PERSON] [13] performed the free vibration analysis to investigate the natural frequencies of an FG rotor-bearing system using the three-dimensional finite element method. Since the FGMs can withstand high temperatures, the influence of temperature gradient on FG structures and rotating systems must be considered.