1. Introduction
Functionally graded materials (FGMs) are an advanced type of composite material that has gained practical importance in recent years. FGMs are composed of metals, ceramics and polymers whose volume fractions vary in the desired directions based on material laws such as power and exponential and sigmoid laws. They are classified into ceramic–ceramic, metal–metal, metal–ceramic, ceramic–polymer and so on. Metal–ceramics are widely used due to their ability to withstand high temperatures, good mechanical performance and high specific strength and fracture toughness. They have been used for rocket engine components, aerospace structures, turbine blades, etc., and have found applications in aerospace, aircraft, automotive, biomedical, power, energy, electronics and chemical industries, among others. The general idea of functionally graded materials was proposed in 1972 for ceramics and polymers, inspired by the material structures of bones, teeth and bamboo trees. The concept of functionally graded materials was developed and the term was coined for the first time by Japanese scientists in the mid-1980s for aerospace applications. They were used as super resistant materials to reduce the generation of thermal stress and improve thermal resistivity in the propulsion systems of spacecraft. Since then, functionally graded materials have gained much importance around the world for various engineering applications [
1,
2,
3,
4,
5]. Given the wide range of applications, FGMs, when used in harsh environmental conditions at elevated temperatures for long durations of time, can undergo corrosion.
Ever since the concept of functionally graded materials was introduced, many researchers have studied their behaviour and performance. Pindera and Dunn [
6] developed a higher-order theory for FG plates subjected to through-thickness thermal gradients and compared the stress fields with finite element (FE) analysis. Aboudi et al. [
7] generalised the higher-order theory based on Cartesian coordinates for FG materials. Recent works on the modelling and isogeometric analysis of functionally graded material structures are reported in the literature [
8,
9]. Few works have been reported in the literature on the vibration analysis of functionally graded beams. Aydogdu and Taskin [
10] studied the free vibration analysis of a simply supported FG beam using Hamilton’s principle for different higher-order shear deformation and classical beam theories. Simsek [
11] analysed the natural frequencies of functionally graded beams for different boundary conditions based on various beam theories. Alshorbagy et al. [
12], using the finite element method, investigated the free vibration characteristics of an FG beam based on the Euler–Bernoulli beam theory.
Functionally graded materials (FGMs) are known for their ability to withstand extremely high-temperature environments. FGMs are also considered potential structural materials for future high-speed spacecraft. Reddy and Chin [
13] studied the dynamic thermoelastic response of functionally graded cylinders and plates using the finite element method, including thermomechanical coupling in the formulation. Kawasaki and Watanabe [
14] simulated a real environment using a H
2/O
2 combustion flame to study the thermal fracture behaviour of FG materials. Lanhe [
15] derived stability and equilibrium equations based on the first shear deformation theory for a functionally graded thick rectangular beam under thermal loads to calculate the buckling temperature.
However, limited works are reported in the literature on the vibration analysis of functionally graded rotor-bearing systems. Gayen and Roy [
16] carried out a vibration and stability analysis of an FG rotor-bearing system using a three-node finite beam element based on Timoshenko beam theory (TBT). Rao and Roy [
17] carried out a dynamic analysis of a functionally graded rotating shaft system using the Timoshenko beam theory. Different analyses have been carried out including the Campbell diagram, stability speed limit and damping ratio. Bose and Sathujoda [
18] performed a natural frequency analysis of a functionally graded rotor system using a three-dimensional finite element model developed using ANSYS (ANSYS 18.0, ANSYS, Canonsburg, PA, USA). Furthermore, Bose and Sathujoda [
19] extended this work and studied the effects of thermal gradients on the vibration characteristics of an FG rotor-bearing system.
The effects of various defects on the vibration characteristics of structures and rotor systems have always been of great significance among researchers, and related works are reported in the literature. Gillichet al. [
20] developed two kinds of mathematical relations for predicting frequency changes due to the two main effects of corrosion—loss of mass and decrease in stiffness. Shekar and Prabhu [
21] studied the effect of coupling misalignments on the vibration characteristics of a rotor-bearing system. Prabhakar et al. [
22] studied a method to detect cracks in a rotor-bearing system by measuring mechanical impedance. However, very limited works are available on the effects of defects present in functionally graded material systems. Gayen et al. [
23] carried out a finite-element-based dynamic analysis of a functionally graded (FG) shaft with a transverse crack using a two-node Timoshenko beam element and considered the effects of translational and rotary inertia, transverse shear deformations and gyroscopic moments. Wattanasakulpong and Ungbhakorn [
24] performed a linear and nonlinear vibration analysis of FG beams with porosities, elastically restrained by the ends, using the differential transformation method. Ferreira et al. [
25] investigated the corrosion behaviour of Al/Al
3Ti and Al/Al
3Zr functionally graded materials formed by the centrifugal casting method and studied the influence of intermetallic platelets on corrosion behaviour. Musbah et al. [
26] investigated the corrosion behaviour of Ti-B4C/CNF functionally graded materials produced in three layers by traditional cold compressing and sintering methods using the potentiodynamic method. Malinina et al. [
27], using a high-velocity oxygen fuel spraying technique, carried out a comparative study of the corrosion resistance of FG (alumina-NiCr) and homogeneous environmental barrier coatings on steel substrates. However, to the best of the authors’ knowledge, works on the effects of corrosion on the vibration characteristics of functionally graded rotor systems are rarely reported in the literature.
Since most of the rotors operate at a constant speed and accelerate or decelerate at a constant angular acceleration or deceleration, uniform corrosion is possible especially when the shaft is exposed to a harsh corrosive environment over a long period at elevated temperatures. In this context, this paper investigates the effects of uniform corrosion on the natural and whirl frequencies of an FG rotor-bearing system using the finite element method based on the Timoshenko beam theory. A parametric study on the effect of corrosion was performed by varying depth, length and position of the corrosion defect. The study was performed for different power law indexes of material distributions in the FG shaft and temperature gradients in the shaft.