Effect of Thermal Gradient on Vibration Characteristics of a Functionally Graded Shaft System

The actual microstructural shapes and gradation of FG material are not available. However, the volume fraction distribution within the FG material can be represented using different mathematical laws. The material properties are varied by varying the distribution of volume fraction with the spatial coordinates. Figure 1 has been taken from the works of Aboudi et al. [21] and shows the continuously graded microstructure of an FG material. For FG shafts, the properties vary radially. A radial variation of volume fraction [17] is shown in Figure 2. The inner core of the shaft purely composed of metal, and the volume fraction of metal decreases, and that of ceramics increases radially outwards. Thus, the outer layer is exclusively made of ceramic, which offers excellent temperature resistance properties. The position dependence is obtained by using the Voigt model, which gives a simple rule for composite materials. The material properties Pi for each layer is expressed as in Eq. (1).

Pi=PcVc+PmVm                (1)

where, P_c and P_m are the material properties of ceramic and metal, respectively, and Vc and Vm refer to the volume fraction of ceramic and metal respectively. The sum of both the volume fractions is unity. Several different mathematical models have been developed over the years by researchers to predict the distribution accurately. Power law, exponential law, Sigmoid law and Mori-Tanka scheme are the laws commonly used for the gradation of FG materials. However, most of the previous works are based on the power-law gradation of FG shafts. In the present study power-law and exponential law gradations have been applied to the rotor systems. Modeling using these laws and their application has been discussed in detail in the subsections. 

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Figure 1. Continuously graded microstructure in cut section of the FG shaft

Apart from radial variation, the properties of FGMs also depend on temperature. In most applications of FG shafts, the outer ceramic is subjected to higher temperatures than the inner metalcore. This difference in gradation generates a radial variation of material properties in the shaft. The temperature-dependent properties have been proposed by Touloukian [22] as in Eq. (2), where P0, P1, P2 and P3 are different material-dependent temperature coefficients. These coefficients are shown in Table 1 and are taken from the work of Reddy et al. [15].

P(T) = P₀ (P_-1 T^-1 + 1 + P₁ T¹ + P₂ T² + P₃T³)           (2)

Stainless Steel-Aluminium oxide (SS-Al₂O₃) is used to model the FG shaft in the present work and the temperature coefficients can be obtained from the works of Reddy et al. [15].

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Figure 2. Radial variation of volume fraction

Table 1. Temperature coefficients for constituent materials

The thermomechanical responses of FGM can accurately be modelled by using different temperature distribution methods obtained from the literature. For an axis-symmetric solid cylinder, assuming no heat generation and solving the 1-D Fourier heat conduction equation with suitable boundary conditions as shown in Eq. (3) yields the radial temperature distribution profiles.

$\frac{d}{d r}\left[r K(r) \frac{d T}{d r}\right]=0$             (3)

2.1 Power law distribution with NLTD

The power law is the most commonly used law to achieve the FG material gradation. It is an established model and has been used extensively to model the behaviour of FG plates, shafts, etc. For a circular cross-section FG shaft or cylinder, the position-dependent material properties P(r) can be expressed as in ref. [15] by Eq. (4): 

$P(r)=P m+(P c-P m) V o(r), V o(r)=\left(\frac{r-R i}{B \rho-B i}\right)^{k}, R i \leq r \leq R o, 0 \leq k \leq \infty$                   (4)

where, P(r) denotes the radially varying material properties such as Young’s Modulus, Poisson’s ratio, Thermal conductivity (K), density, and coefficient of thermal expansion. Pm and Pc are temperature-dependent material properties at the ceramic rich and metal-rich region, respectively. These are obtained by solving Eq. (3) at respective metal and ceramic temperatures. Ro and Ri are the outer and inner radii of the cylinder, respectively. ' k' is the power-law index. The volume fraction of metal, at a mid-radius 'r 'of any layer is represented by Vo(r).

NLTD has been studied in several research works on FGMs and is used with power-law gradation. It is the solution to Eq. (3) subjected to mentioned boundary conditions and considering the first seven terms of the polynomial expansion [23] gives Eq. (5):

$T(r)=T m+(T c-T m)\left[\sum_{j=0}^{5}\left\{\frac{(-1)^{j}}{j k+1}\left(\frac{K c m}{K m}\right)^{j}\left(\frac{r-R i}{R o-R i}\right)^{j k+1}\right\}\right] /\left[\sum_{j=0}^{5} \frac{(-1)^{j}}{j k+1}\left(\frac{K c m}{K m}\right)^{j}\right]$                 (5)

where, Kcm = Kc – Km. Kc and Km refer to the thermal conductivity of ceramic rich and metal-rich regions at a given temperature. Kc and Km are temperature dependent and are also calculated using Eq. (4) at respective metal and ceramic temperatures.

2.2 Modelling using exponential law gradation with ETD

Several researchers have used exponential law for FG plates, disks and shells, but works on FG shafts are very scarce. The position-dependent material properties using exponential law for a circular cross-section FG shaft can be given as [24] shown in Eq. (6):

$P(r)=\operatorname{Pm} \exp \{\lambda(r-R i)\}, \quad$ where $\lambda=\left(\frac{\ln \left(\frac{P c}{P m}\right)}{R o-R i}\right), \quad R i \leq r \leq R o$            (6)

P(r) denotes the position-dependent material properties. Pc and Pm are properties of the ceramic rich and metal-rich region, respectively. This law governs all the material property distributions except for Poissons’s ratio. Poissons’s ratio is kept constant under exponential law.

Exponential temperature distribution (ETD) is used to estimate the temperature distribution profile with exponential law. The ETD is obtained using the thermal conductivity equation for cylinders with appropriate boundary conditions. Eq. (7) can be used to obtain the temperature distribution along cross-section 0f the shaft [10].

$T(r)=A+B e^{-\lambda(r-R i)}, \quad R i \leq r \leq R o$              (7)

where,

$A=T m-\frac{(T c-T m)}{e^{-\lambda(R o-R i)-1}}, B=\frac{(T c-T m)}{e^{-\lambda(R o-R i)-1}}, \lambda=\left(\frac{\ln \left(\frac{K c}{K m}\right)}{R o-R i}\right)$

T(r) is the temperature value at radial distance 'r', and Tc and Tm are temperatures at the ceramic rich and metal-rich regions. Pm and Pc refer to the position-dependent material properties at the ceramic rich and metal-rich area, respectively. The following temperature distribution law can be used for the analysis of shafts, where exponential gradation is desired.

2.3 Modelling in present work

The FG rotor is composed of SS-Al₂O_3. The mechanical properties of the individual constituents of these FG materials have been shown in Table 2. The FG rotor is divided into twenty layers radially to estimate the continuous material distribution. The radial division is selected based on convergence study during validation with literature, however more number of layers can be assigned by the code based on computational ability. A Python code has been developed to generate ANSYS macros that contain the desired material distribution. The user can give input for the dimensions of the rotor, choice of material distribution law, temperature distribution and the size of each layer. Using this information, the material properties are assigned to each segment by calculating its value at the layer mid-radius. The output is generated in the form of macros, which can be imported in ANSYS to achieve the gradation. Figure 3 shows the layer-wise material variation present in the SS-Al₂O₃ FG shaft modelled using ANSYS.

Table 2. Mechanical properties of constituent materials of FGMs at different temperatures

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Figure 3. Material gradation in SS-Al₂O₃ shaft modelled using ANSYS

However, for more accurate assignment of the density, each layer is assigned the layer average density. The density function is taken to vary as power-law or exponential law and is averaged over the inner radius to the outer radius of the layer. The equations Eq. (8) and Eq. (9) show the averaged density equations for power law and exponential law, respectively, where $\rho c$ and $\rho m$ are the densities of metal and ceramic used. These equations give a more uniform density variation than directly applying mid-radius density to the layer. Here, r1 and r2 refer to the internal and external radius of the selected layer.

$\rho \mathrm{avg}=\frac{\int_{r 1}^{r 2}\left[(\rho \mathrm{c}-\rho \mathrm{m})\left(\frac{r-R i}{R o-R i}\right)^{k}+\rho \mathrm{m}\right] d r}{\int_{r 1}^{r 2} d r}$                     (8)

$\rho \operatorname{avg}=\frac{\int_{r_{1}}^{r_{2}}[(\rho m) \exp \{\lambda(r-R i)\}] d r}{\int_{r_{1}}^{r_{2}} d r},$ where $\lambda=\left(\frac{\ln \left(\frac{\rho c}{\rho m}\right)}{R o-R i}\right)$                (9)

The equation of motion for the complete rotor-bearing system can be expressed as:

$M \ddot{p}(\mathrm{t})-\Omega \mathrm{G} \dot{p}(\mathrm{t})+K p(t)=f(t)$                 (10)

where, M is the global mass matrix, including the rotary and translational masses of the shaft, G is the global gyroscopic matrix, K is the global stiffness matrix, including stiffness of the shaft and the bearing. p(t) is the global nodal displacement vector, and f(t) is the global external force vector. For the free vibration analysis, external and gravity forces are neglected.  

Block Lanczos eigenvalue extraction method available in ANSYS has been used for obtaining the natural frequencies of the simply supported shaft. The modal analysis has been performed using the QR damped solver of ANSYS [26]. Following is the coordinate transformation (Eq. (11)) used to transform the complete eigenvalue problem into modal subspace:

$\{p(t)\}=[\phi]\{y\}$                 (11)

$[\Phi]$ refers to the normalized eigenvector matrix for the mass matrix [M]

$\{\mathrm{y}\}$ is the vector of modal coordinates

Using transformation Eq. (11) in Eq. (10) for no external forces, the differential equation of motion in modal subspace can be written as given in Eq. (12).

$[I]\{\ddot{y}\}+[\phi]^{T}[C][\phi]\{\dot{y}\}+\left(\left[\Lambda^{2}\right]+\right.\left.\quad[\phi]^{T}[\text {Kunsym}][\phi]\right)\{y\}=\{0\}$               (12)

where, $\left[\Lambda^{2}\right]$ is a diagonal matrix containing the first n eigenfrequencies $\omega_{i}$_.

A double disc FG rotor bearing system, shown in Figure 4 is composed of SS-Al₂O₃, is considered to investigate the natural frequencies and whirl effects.  Modelling has been done involving both power and exponential law. The effect of material gradation on whirl frequencies has been studied. Campbell diagrams have been plotted to capture the backward and forward whirl frequencies. Attempts have been made to identify and present the trends in the variation of frequencies.

6.1 Effect of material gradation on natural frequencies at a constant temperature

Subsections 6.1.1 and 6.1.2 discuss effect of power law coefficients on natural frequency of FG shafts modelled using power law and exponential law respectively.

6.1.1 FG shaft system graded using power law without thermal gradient

The FG rotor bearing system shown in Figure 4 is considered. The radial variation of material properties has been achieved using power law gradation. The shaft is assumed to be at a constant temperature of 300K. Six power-law coefficients ranging from 0.3 to 10 are selected to investigate the effect on frequencies with a change in coefficients. Each coefficient generates a unique set of radially varying material properties. The first six whirl natural frequencies at 4000 rpm rotating speeds of the shaft has been presented in Table 5.

Table 5. Whirl frequencies (Hz) of SSAl₂O₃ FG shaft at 4000 rpm

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Figure 9. Natural frequency variation for SS-Al₂O₃ FG rotor system modelled using power law at room temperature

The natural frequencies of the shaft systems at stationary condition show a decreasing trend with an increase in the value of power-law coefficients. The decrease in frequencies occurs in a non-linear fashion. The variation of natural frequencies with power law index is shown in Figure 9. The curve clearly illustrates that the slope of the curve decreases gradually. This shows that the decrease in natural frequencies occur at a slower rate for a higher power-law coefficient. The reason for this behavior can be attributed to the variation of average Young’s modulus for the FG shaft with power law coefficients (Figure 10). The plot exhibits that at lower values of power law index there is a steep drop in average stiffness of the FG shaft, however as the power law coefficients increase the drop becomes gradual.

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Figure 10. Variation in average Young’s modulus for SS-Al2O3 FG rotor system with power law index at room temperature

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Figure 11. First BW/FW frequencies at 4000 rpm for SS-Al₂O₃ FG shaft without thermal

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Figure 12. Campbell diagram for FG rotor system modelled using power law (k=0.3)

Another trend in whirl frequencies is observed at a shaft rotation speed of 4000rpm. The percentage of the separation in frequencies at 4000 rpm with respect to stationary shaft frequencies has been investigated. The percentage variation in whirl frequencies is calculated using Eq. (13). Where $\omega_{4000}$ and $\omega_{0}$ represent forward/backward whirl frequency at 4000 rpm and 0 rpm respectively.

$Percentage variation in whirl frequency=\frac{\omega_{4000}-\omega_{0}}{\omega_{0}} x 100 \%$            (13)

The trends of the percentage differences have been represented graphically shown in Figure 11 The separation of both first backward (1BW) and first forward whirl (1FW) frequencies have been examined. Both are found to increase with the rise in power-law coefficients. It can also be noted that the percentage difference of 1^st FW frequency is less than 1^st BW frequency at any k value. This behavior is due to the variation pattern of Young’s Modulus for the shaft system shown in Figure 10. Figure 12 shows the Campbell diagram for the SS-Al₂O₃ FG rotor system modelled using power law with power law coefficient as 0.3.

6.1.2 FG shaft system graded using exponential law without thermal gradient

Results are generated similar to previous system, however the exponential gradation law is used to model the shaft system. The material properties such as Young’s modulus are varied using the exponential gradation but Poison’s ratio is taken as a constant. The average value of Poisson’s ratio of the metal and ceramic has been applied to the shafts. The whirl frequencies have been computed at the stationery condition and at a speed of 4000rpm. The first three forward and backward whirl frequencies have been shown in Table 5.

The results show that the natural frequencies obtained for SS-Al₂O₃ shaft frequency at power-law coefficient one corresponds to the frequencies obtained using exponential gradation. This nature can be expected as the material property distribution curve for exponential law Figure 7 are similar to the corresponding power law coefficient curve of material gradation, as shown in Figure 6. Due to gyroscopic effects of the shaft and the discs, separation of natural frequencies occurs at higher shaft speeds. Figure 13 represents the Campbell diagram obtained for the exponentially graded shaft system

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Figure 13. Campbell diagram for FG rotor system modelled using exponential law

6.2 Effect of thermal gradient on whirl frequencies

To effects of thermal gradient on whirl frequencies of FG rotor systems are presented in this section. The SS-Al₂O₃ FG shaft as shown in Figure 4 is used for the analysis. Results have been presented for different thermal gradients and the reasons for the variations obtained have been investigated. The inner core metal temperature (Tm) and the outer ceramic temperature (Tc) of the shaft are varied to apply the effect of thermal gradation. The ceramic temperature is kept higher to replicate the general application conditions of FG material applications. The effects of booth NLTD and ETD have been discussed.

6.2.1 Shaft modelled using power law and NLTD

The whirl frequencies of the shaft system subjected to thermal gradients has been studied in this section. The shaft has been subjected to four different temperature gradients with different metal core (Tm) and outer ceramic temperatures (Tc). Table 6. shows the effect of temperature gradation (∆T) on first natural frequencies. The table also presents the effect of increasing metal core temperature (Tm) for the same temperature gradients on natural frequencies.

Table 6. Natural frequency comparison for different rotor system subjected to thermal gradients

The trends obtained in the variation of these frequencies have been represented graphically in figures. In Figure 14, a reduction in natural frequencies occurs with an increase in temperature gradients for all metal core temperatures is observed. However, it is worth noting that this reduction is more prominent at lower k values. At higher k values, the reduction in natural frequencies is very less for an increase in a temperature gradient. To verify that the trend obtained is due to the FG shaft and does not depend on the effects due to the two discs, an FG shaft composed of SS-Al₂O₃ with same dimensions and boundary conditions but without the two discs has been modelled analysed. The inner metal core temperature (Tm) is kept at 300K. The trends in the variation of natural frequency for shafts made using different power law coefficients subjected to the same temperature gradients (∆T) have been represented graphically in Figure 15. Significant similarities in the trend can be observed between Figure 14 and Figure 15, validating that the characteristics of natural frequency variation is applicable to the FG shaft systems.

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Figure 14. Natural frequency variation for FG shaft with disc

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Figure 15. Natural frequency variation for FG shaft without disc

The reason for this behavior can be clearly seen in Figure 17, which represents the average elastic modulus of the FG shaft modelled using different k values with a change in thermal gradients. The Young’s modulus decreases with increase in temperature gradient for a shaft made by keeping power law coefficient as 0.3. However, the drop in average Young’s modulus of FG shaft having higher power-law coefficients (k=5 and k=10) is almost negligible. Thus, the lower change in elastic modulus corresponds to the small change in natural frequencies at higher power-law coefficients.

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Figure 16. Variation of first natural frequencies with Tm for k = 0.3 and k = 10

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Figure 17. Variation of average Young’s Modulus with temperature gradient for different power-law index (k)

Another significant observation is that at any power-law coefficient as the metal core temperature is increased, the natural frequencies keep decreasing. The reason for such trends is due to a reduction in Young’s modulus or stiffness of the shaft with increasing temperature. This variation in natural frequencies has been presented in Figure 16 for SS-Al₂O₃ shaft modelled using k=0.3. and k=10.

The effects of thermal gradient on whirl frequencies of the disc rotor systems have also been investigated. The shaft rotation speed is taken as 4000 rpm, and the separation of frequency pairs produces forward and backward whirl. The results show that the whirl frequencies also follow the general trend, i.e. their values decrease with the increase of temperature gradient. The splitting of both BW and FW frequency increase with the rise in temperature gradient for SS-Al₂O₃ shaft which is evident from the Campbell diagrams shown in Figure 18.

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Figure 18. Campbell diagram- k=5, NLTD Tm=600K, Tc=900K

6.2.2 Shaft modelled using exponential law and ETD

The effect of the thermal gradient on natural frequencies has been examined for shaft system modelled using exponential law. The temperature profile has been generated using the Exponential temperature distribution method. The trends obtained in the variation of these frequencies are shown in Figure 19. It can be noted that the natural frequencies decrease as expected, but the decrease is more significant when compared to shafts modelled using power law having same temperature gradients. Percentage decrease in the natural frequencies of SS-Al₂O₃ shaft modelled using power law for k value of 0.3 with metal-core temperature 300K is 1.22%. Whereas the percentage decrease is 2.25%. for exponentially graded shaft having the same geometry and with same metal-core temperature.

Further as shown in Figure 19, for a power law graded FG shaft, the rate of decrease in first natural frequencies decreases with a rise metal-core temperature (Tm). However, for an exponentially graded FG shaft, the rate of decrease in natural frequencies rises with the increase in metal core temperature (Tm).

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Figure 19. Natural frequency variation for exponentially graded FG shaft under thermal gradient

The reasons for the variations can be attributed to the variation of Young’s modulus of the shaft system with temperature. Figure 20 shows the variation of average Young’s modulus with increasing temperature gradients. The upper portion of the graph represents the FG (SS-Al₂O₃) shafts modelled using power law (k=0.3) and NLTD while the lower portion refers to shafts following exponential gradation and ETD. The plot clearly shows that average Young’s modulus decreases at decreasing rate with increasing temperature gradient when the power law is used to model but it falls at an increasing rate with an increase in temperature gradient for exponentially graded shafts. The difference in average elastic modulus becomes more significant with increase in temperature gradient for exponentially graded FG shafts.

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Figure 20. Average Young’s modulus variation for exponentially graded FG shaft under thermal gradient

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Figure 21. Campbell diagram (ETD, Tm=300K, Tc= 600K)

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Figure 22. Campbell diagram (ETD, Tm=600K, Tc= 900K)

The first three FW and BW frequencies for the shaft system, rotating at a speed of 4000 rpm, subjected to different temperature gradients are analysed. The Campbell diagrams have been plotted for shaft systems subjected to thermal gradients shown in Figure 21 and Figure 22. The splitting of whirl frequencies increases with the increase in the temperature gradients and metallic core temperature. The reason for this behavior is due to the rate of variation of average Young’s modulus for the FG shaft at different temperature gradients. Table 6 depicts the first natural frequency of different rotor systems subjected to thermal gradients. The advantage of using FG shafts at higher temperature can be inferred from the results. FG shafts graded using power law and exponential law show small difference in frequencies at higher temperatures. However, for a homogenous stainless-steel shaft, the fall in natural frequencies is significant. The reason for this the abrupt decline in Young’s modulus of steel at higher temperatures (above 900K).