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 (FGMs) 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]. Simsek [
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. Aydogdu and Taskin [
6] analysed the natural frequencies of the FG beam based on exponential and parabolic shear deformation beam theories. Nguyen et al. [
7], based on the first-order shear deformation theory, investigated the static and free vibration of an axially loaded FG beam. Pradhan and Chakraverty [
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 Simsek [
9] based on classical and different higher-order shear deformation beam theories. Azadi [
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. Nelson and McVaugh [
11] generalised Ruhl’s element by including the effects of gyroscopic moments and rotatory inertia. By including the transverse shear effects, Nelson [
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. Bose and Sathujoda [
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. Kiani and Eslami [
14] performed the buckling analysis on FG beams under different types of thermal loading. Mahi 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 Bose and Sathujoda [
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. Dimarogonas [
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 Mayes and Davies [
18]. Using the Paris’s equations, Dimarogonas and Papadopoulos [
19] developed a local flexibility matrix for a cylindrical shaft having a crack. Papadopoulos [
20] studied the torsional vibrations of a rotor having a transverse crack. Papadopoulos and Dimarogonas [
21] developed a compliance matrix by coupling the bending and torsional vibration of a Timoshenko rotor with a transverse crack. Darpe 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. Ichimonji and Watanabe [
23] analysed the transverse vibration of a slant cracked rotor system by considering the breathing effect of the crack. Dias-da-Costa et al. [
24] presented a novel image deformation approach to monitor the propagation of the crack. Yao et al. [
25] discussed the various characterised methods to detect the cracks. Sekhar and Prasad [
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, Sekhar et al. [
27] detected and monitored the slant crack of the rotor system. Prabhakar et al. [
28] used continuous wavelet transform to detect and monitor the cracks in rotors. Prabhakar 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]. Gayen 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. Bose 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. To the best of the author’s knowledge, the steady-state and transient responses of an exponential-law based slant-cracked functionally graded rotor-bearing system to detect and understand the influence of the cracks have not been reported in the literature. Therefore, the principal aim of the present work is to study the natural frequencies, steady-state and transient time responses, and the corresponding frequency spectra of a slant cracked exponentially graded (EG) rotor-bearing system by considering the breathing effect of the slant crack.
The organisation of this current research work is as follows. Primarily, an FG rotor-bearing system having an exponentially graded shaft is considered in the present work. The material properties are graded along the radial direction of the FG shaft by using exponential law, whereas exponential temperature distribution (ETD) law is used to simulate the temperature gradients across the cross-section of the FG shaft. Although Sekhar and Prasad modelled a slant crack in a steel rotor-bearing system [
26], the derivations and modelling of the slant-cracked EG rotor-bearing system are not available in the literature. Therefore, a two-nodded EG rotor element with and without a slant crack are formulated based on Timoshenko beam theory using the finite element method. Then, the stiffness matrix of a slant-cracked EG rotor element is developed by determining the local flexibility coefficients using Paris’s equations. Transient and steady state vibration responses of EG rotor system have been simulated by considering the breathing slant crack in order to develop a crack detection method through frequency response. Finally, the influence of the crack parameters and temperature gradient on natural frequencies are investigated, and the effect of the crack depth, torsional frequency and temperature gradient on steady-state and transient responses of a slant-cracked EG rotor system is also analysed.
2. Materials and Methods
Material properties of an FGM vary along a particular direction known as gradation direction. The radial direction is considered as gradation direction in the exponentially graded (EG) shafts with a circular cross-section. Therefore, material constituents of the shaft are distributed along the radial direction of the shaft. The EG shaft is composed of ceramic constituents on the outer layer of the shaft, and the metallic components are present in the inner core of the EG shaft. The outer ceramic layer is made of Zirconia (ZrO
2), whereas the inner metallic core is composed of Stainless Steel (SS). The percentage of the ceramic constituents increases, and the rate of metal constituents decreases while moving across the cross-section from the inner core of the EG shaft. Therefore, the material properties such as Young’s modulus and Poisson’s ratio depends on the position of material constituents. The position of the material properties is determined by the Voigt model [
35], which is a simple rule for mixtures of composites. The material properties of a specific layer
is given as
where
and
are volume fractions of metal and ceramic, respectively, whereas
and
are material properties of metal and ceramic, respectively. The sum of volume fractions of metal and ceramic is
Since the material properties vary with the temperature, the temperature dependency of the material properties is achieved by using Equation (3) [
36].
T is the temperature in Kelvin.
,
,
,
and
are the coefficients of the temperature, which are listed by Reddy and Chin [
37]. Since the volume fractions of the material constituents are difficult to determine, the material gradation laws are used. Exponential law is employed in the present work, and its detailed explanation is presented in the following subsection.
2.1. Material Gradation Based on Exponential Law
Material gradation laws such as exponential law, power-law and sigmoid law are used to assign the material properties to an FG shaft. Exponential law is used in the present work as it has an easily estimated parameter (
λ) and mathematically controllable. Based on exponential law, the material properties can be expressed as
is the temperature and position dependent material property.
and
are the outer and inner radius of the FG shaft, respectively. In the present work, an EG shaft with an inner radius
= 0 is considered as shown in
Figure 1.
2.2. Exponential Temperature Distribution
Temperature is also assumed to vary across the cross-section of the EG shaft. Normally, 1D Fourier heat conduction equation is used to determine the temperature variation along the solid circular shaft whose inner and outer radius are
and
, respectively.
is the thermal conductivity.
r is the radial distance.
and
are inner and outer temperatures of the shaft. Heat generation is not considered in the present work. For boundary conditions,
.
where
,
and
A python code has been developed to assign the material properties and distribute the temperature across the cross-section of the cracked EG shaft.
8. Conclusions
The dynamic analysis of a slant-cracked EG rotor-bearing system has been carried out using the finite element method to investigate the influence of a slant crack on natural frequencies, steady-state and transient responses. Exponential law is used to grade the material properties along the radial direction of an EG shaft, and exponential temperature distribution law is used for temperature gradation across the cross-section. The effect of crack depth, thermal gradient, crack location, torsional frequency, rotor speed, and angular acceleration on eigenfrequencies and dynamic responses have been investigated, and the following important conclusions are drawn from the analysis.
The decrease in natural frequencies with an increase in crack depth is significant when a crack is located near the disc or far from the bearings. The stiffness of the cracked EG rotor is reduced due to the presence of a slant crack.
The percentage decrease in the natural frequency is less when a slant crack is present near the bearings compared to the percentage decrease in natural frequency when a slant crack is near the disc. The reduction in stiffness of the EG shaft is compensated by the bearing stiffness when a crack is present near the bearings. Therefore, the natural frequencies of an EG rotor-bearing system are hardly affected when a slant crack is present near the bearings.
Natural frequencies of the EG rotor-bearing system decrease with the increase in the crack depth as the stiffness is affected. The increase in temperature gradient of the EG rotor decreases the stiffness as the Young’s modulus is affected. Therefore, the stiffness of a cracked EG rotor-bearing system is furthermore reduced due to an increase in the thermal gradients. The sudden drop in the natural frequencies for different crack depths at higher thermal gradients has also been observed for this reason.
Subharmonic peaks are found to be centred on the operational speed at an interval frequency corresponding to the torsional crack breathing frequency in steady-state frequency spectra of a slant-cracked EG rotor-bearing system. The amplitude of the harmonic peaks is increased with an increase in crack depth. Hence the severity of the crack could be identified through the sub-harmonic peaks.
The frequency spectrum of the transient response of a slant-cracked EG rotor-bearing system is found to have the subharmonic frequencies centred on the critical speed of the rotor system at an interval frequency corresponding to the torsional frequency corresponds to crack breathing frequency. The presence of subharmonics on the frequency spectra confirms the presence of the crack in the EG rotor.
Since a slant crack may develop due to excessive torsional vibrations of rotor systems, which is possible during the start-up and shutdown of machines, it is essential to detect the crack through vibration response to avoid catastrophic failures. From the present study, a slant crack could be detected through steady state and transient responses of an EG rotor system by extracting the subharmonic peaks corresponding to crack breathing frequency. However, for crack depths below 0.1, other detection techniques need to be explored. The work presented in this paper is based on the theoretical validations, and experiments are being planned for the near future.