Free Vibration Analysis of a Thermally Loaded Porous Functionally Graded Rotor–Bearing System

Abstract

The present work deals with natural and whirl frequency analysis of a porous functionally graded (FG) rotor–bearing system using the finite element method (FEM). Stiffness, mass and gyroscopic matrices are derived for porous and non-porous FG shafts by developing a novel two-noded porous FG shaft element using Timoshenko beam theory (TBT), considering the effects of translational inertia, rotatory inertia, gyroscopic moments and shear deformation. A functionally graded shaft whose inner core is comprised of stainless steel (SS) and an outer layer made of ceramic (ZrO2) is considered. The effects of porosity on the volume fractions and the material properties are modelled using a porosity index. The non-linear temperature distribution (NLTD) method based on the Fourier law of heat conduction is used for the temperature distribution in the radial direction. The natural and whirl frequencies of the porous and non-porous FG rotor systems have been computed for different power law indices, volume fractions of porosity and thermal gradients to investigate the influence of porosity on fundamental frequencies. It has been found that the power law index, volume fraction of porosity and thermal gradient have a significant influence on the natural and whirl frequencies of the FG rotor–bearing system.

Dynamic Stiffness Matrix Approach to Free Vibration Analysis of Functionally Graded Rotor Bearing System Subjected to Thermal Gradients

“…, 8) are the constants of integration. The two sets of integration constants (C 1 to C 4 ) and (C 5 to C 8 ) in Equations ( 28) and (29), respectively are related to each other due to the coupling between the lateral deflection and the bending slope in both (x,s) and (y,s) planes given by the coupled equilibrium Equations ( 12), ( 13), ( 17)and (18), respectively. The relationship between the two sets of integration constants has been derived.…”

“…The metal and ceramic materials considered for the material modeling of the FG rotor are stainless steel and zirconium dioxide, respectively. The temperature dependent coefficients for the material properties of stainless steel and zirconium dioxide are obtained from [18] and are tabulated in Table 3 The first three modes of the forward and backward whirl frequencies of the FG rotor bearing system subjected to temperature gradients at the rotor spin speed 4000 rpm and inner temperature T i = 300 K has been computed using the dynamic stiffness method for different power law indices and temperature gradients. The computed whirl frequencies of the FG rotor bearing system subjected to temperature gradients havebeen compared with the FEM results available in [18].…”

“…The temperature dependent coefficients for the material properties of stainless steel and zirconium dioxide are obtained from [18] and are tabulated in Table 3 The first three modes of the forward and backward whirl frequencies of the FG rotor bearing system subjected to temperature gradients at the rotor spin speed 4000 rpm and inner temperature T i = 300 K has been computed using the dynamic stiffness method for different power law indices and temperature gradients. The computed whirl frequencies of the FG rotor bearing system subjected to temperature gradients havebeen compared with the FEM results available in [18]. The comparison between the results from the DSM method in the paper and FEM in the literature for temperature gradients at ∆T = 0 K, 300 K and 600 K are presented in Tables 4-6.…”

“…The free vibration analysis of the FG rotor bearing system having hysteresis and viscous damping effects has been carried out in refs [15,16] using FEM. The free vibration behavior of thermally loaded FG rotor-bearing systems having defects, such as transverse crack [17], porosities [18] and corrosion defect [19] have been investigated, and the influence of these defects on the natural and whirl frequencies has been studied in detail in the literature.…”

Steady State and Transient Vibration Analysis of an Exponentially Graded Rotor Bearing System Having a Slant Crack

“…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.…”