## 1 Introduction
[PERSON] and [PERSON] [14] performed the buckling analysis on FG beams under different types of thermal loading. [PERSON] et al. [15] investigated the free vibration frequencies of an FG beam for different material distributions (power-law, exponential-law and sigmoid-law) under a thermal environment based on a unified higher-order shear deformation theory. The effect of temperature gradients on fundamental frequencies of an FG rotor-bearing system based on the Timoshenko beam theory is investigated by [PERSON] and [PERSON] [16]. Cracks are formed in the mechanical components of rotating machinery when they are subjected to excessive fatigue loads during the operation. Therefore, the dynamic response of the rotor systems had been analysed by several researchers. [PERSON] [17] had observed that due to the presence of a crack, the local flexibility of the rotor is affected and developed an analytical formulation of local flexibility in relation to depth to investigate the dynamic response of the rotor under the influence of a crack. The flexural vibration behaviour of a transverse-cracked rotor-bearing system with multiple rotors is studied by [PERSON] and [PERSON] [18]. Using the Paris's equations, [PERSON] and [PERSON] [19] developed a local flexibility matrix for a cylindrical shaft having a crack. [PERSON] [20] studied the torsional vibrations of a rotor having a transverse crack. [PERSON] and [PERSON] [21] developed a compliance matrix by coupling the bending and torsional vibration of a Timoshenko rotor with a transverse crack. [PERSON] et al. [22] computed the steady-state unbalance response of a cracked Jeffcott rotor system subjected to periodic axial impulses using the Runge-Kutta method. [PERSON] and [PERSON] [23] analysed the transverse vibration of a slant cracked rotor system by considering the breathing effect of the crack. [PERSON] et al. [24] presented a novel image deformation approach to monitor the propagation of the crack. [PERSON] et al. [25] discussed the various characterised methods to detect the cracks. [PERSON] and [PERSON] [26] formulated the local flexibility coefficients for a slant-cracked rotor element and studied the steady-state response of a rotor-bearing system having a slant crack using the finite element method. By using mechanical impedance, [PERSON] et al. [27] detected and monitored the slant crack of the rotor system. [PERSON] et al. [28] used continuous wavelet transform to detect and monitor the cracks in rotors. [PERSON] et al. [29] studied the transient responses of a rotor system having a slant crack passing through the critical speeds by applying an unbalanced force and harmonically varying torque on the rotor. While the above literature review divulges that the various researchers had been investigating the vibration response of cracked homogeneous rotors, analysis of the vibration behaviour of the cracked FG rotors is limited. Only a few works were reported on studying the dynamic response of the FG structures and rotors having defects in the literature. Dynamic analysis has been performed on porous and corroded FG rotor-bearing [30; 31]. [PERSON] et al. [32; 33] investigated the whirl and natural frequencies of a transverse cracked functionally graded rotor-bearing system. A slant crack might develop when an FG rotor is subjected to excessive torsional vibrations. Hence, it is significant to study the dynamic behaviour of an FG rotor-bearing system having a slant crack. [PERSON] et al. [34] performed the whirl frequency analysis on an FG rotor-bearing system having a slant crack. Although a few researchers performed the dynamic analysis to investigate the natural and whirl frequencies of the power-law based FG rotors with defects (porosity, corrosion, transverse crack and slant crack), detecting the presence of crack and its influence on the dynamic responses of an FG rotor have not been reported in the literature yet.