# Free Vibration Analysis of a Thermally Loaded Porous Functionally Graded Rotor-Bearing System
## 7 Conclusions  
A dynamic analysis of a porous functionally graded rotor-bearing system has been carried out using the finite element method. The stiffness and mass matrices of a two-noded porous FG shaft are derived in this work. The FG rotor shaft's inner core is comprised of stainless steel and its outer layer is made of ceramic ZrO\({}_{2}\). The effects of porosity, power law index and thermal gradients on the natural and whirl frequencies of an FG rotor system have been analysed and the following conclusions can be drawn from the analysis:  
* DNFs decrease with an increase in the volume fraction of porosity (\(\alpha\)) due to the reductions in the material properties, which lead to a drop in the stiffness and mass of the porous FG shaft.
* DNFs of the FG shaft with low volume fraction porosity are slightly higher than the DNFs of the FG shaft without porosity. The drop in stiffness and mass escalated the DNFs of the porous FG shaft.
* DNFs of the FG rotor system initially decrease with the increase in k, but later the rate of decrease in DNFs is comparatively low. This is because the percentage of metal is significantly higher at higher power law indices.
* DNFs decrease with an increase in thermal gradients--the reason being that Young's Modulus reduces at elevated temperatures, which, in turn, reduces the stiffness of the system. However,  
Figure 10: (a) Campbell diagram of an FG rotor–bearing system for k = 0.5 and \(\Delta\)T = 0, without porosity. (b) Campbell diagram of a porous FG rotor–bearing system for k = 0.5, \(\alpha\) = 0 and \(\Delta\)T = 600. (c) Campbell diagram of a porous FG rotor–bearing system for k = 0.5, \(\alpha\) = 0.1 and \(\Delta\)T = 0. (d) Campbell diagram of a porous FG rotor–bearing system for k = 0.5, \(\alpha\) = 0.1 and \(\Delta\)T = 600.  
due to the presence of porosities in the FG shaft, the DNFs drop abruptly at low values of k; however, later, the growth is hardly detectable.
* Whirl frequencies decrease with an increase in the power law index and thermal gradients. The split between the second forward and backward whirl frequencies is significant compared to the split between the first and third whirls.  
As the natural and whirl frequencies are affected by various factors of a porous FG rotor system, such as the power law index, volume fraction of porosity and thermal gradients, it is important to analyse the effects of these factors to accurately predict the critical speeds of an FG rotor-bearing system in order to avoid any rotor failures due to resonance. The present analysis provides an important insight into the natural frequency analysis of a porous FG rotor system, which can be used to analyse the dynamic behaviour of a porous FG rotor system at its critical speeds. Further work in this direction could include the modelling of uneven porosity distributions in an FG rotor system in order to investigate their effects on the dynamic properties of the system.  
Conceptualisation, P.S.; methodology and validation, P.S., A.B. and B.O.; writing--original draft preparation, A.B. and B.O.; writing--review and editing, P.S., G.C., R.C. and A.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The authors declare no conflict of interest.