Finite Element Analysis for Dynamic Response of Rotor-Bearing System With Cracked Functionally Graded Turbine Shaft

This paper describes finite element (FE) analysis of a rotor-bearing system having a functionally graded (FG) shaft with a transverse breathing crack. Two nodded Timoshenko beam element with four degrees of freedom (DOFs) per node has been considered and effects of translational and rotary inertia, transverse shear deformations, and gyroscopic moments are also considered. The FG shaft is considered to be composed of zirconia (ZrO₂) and stainless steel (SS) with the volume fraction of SS increasing towards the inner radius of the shaft. Thermo-elastic material properties are considered in the radial direction of the FG shaft following power law gradation. Local flexibility coefficients (LFCs) of the cracked FG shaft are determined analytically as a function of crack size, power law gradient index (k), and temperature with crack orientation using the Castigliano’s theorem and Paris’s equations which are used to compute the stiffness matrix in the FE analysis. The FE formulation has been validated with the analytical and FE solutions reported in the literatures, and then natural frequencies and whirling (forward and backward) frequencies are determined. Influences of crack size, power law gradient index, slenderness ratio, and temperature gradient with crack orientation, on the dynamic responses of the rotor-bearing system with an FG shaft are studied. Results show that the power law gradient index has significant influence on the natural frequencies and whirling frequencies for the rotor-bearing system with the breathing cracked FG shaft and the choice of power law index could play an important role in design of FG shafts under thermo-mechanical environment from the view point of damage tolerant design.