Transverse vibration and stability of a functionally graded rotating annular disk with a circumferential crack

Abstract

Out of plane vibration of rotating disks limits their performances especially at certain critical speed. The critical speed of these disks may be affected by the presence of defects such as a circumferential crack. In this paper out of plane vibration of functionally graded (FG) rotating annular disks with a circumferential open crack is investigated. The cracked disk is modeled as two sub-disks, connected at the crack location by translational and rotational line springs, simulating the crack plate response to induced shear force and bending moment at the crack radius. These spring stiffness constants are obtained numerically using the finite element method (FEM) as a function of crack depth and radius. The rotational spring stiffness strongly depends on the disk rotation speed, while the stiffness of the translational spring is found to be independent of the disk speed. Both spring constants depend on the spatial distribution of the disk elastic modulus.The in-plane disk stresses are obtained using a semi-analytical approach. Those in plane stresses are used to obtain the governing equation of out of plane motion of the disk. A finite difference scheme is used to solve the partial differential equation of motion to obtain eigenvalues, critical speed and associated mode shapes. The lowest critical speed, which is one of the important parameters limiting the performance of the rotating disk, is obtained from the Campbell Diagram. It is found that irrespective of the distribution of the modulus of elasticity in the FG disk, increasing the crack depth or decreasing the crack radial distance from the disk center decreases the critical speed. The critical speed reduction is more pronounced for the case when the disk material modulus of elasticity is decreasing from the disk center.