Triply coupled bending–torsion vibration of Timoshenko and Euler–Bernoulli shaft beams with arbitrarily oriented open crack

Abstract

The paper presents the full formulation for a crack model for analyzing the triply coupled free vibration of both Timoshenko (short) and Euler–Bernoulli (long) shaft beams based on compliance approach in the presence of a planar open edge crack in an arbitrary angular orientation with a reference direction. The compliance coefficients to account for the local flexibility due to the crack for both the beams have been obtained through the concept of strain energy release rate and crack tip stress field given in terms of the stress intensity factors. The type of disturbance in stress–strain field that a continuous cracked beam theory can accommodate is not within the scope of the model. The compliance matrices for the Timoshenko (short) and Euler–Bernoulli (long) beams, respectively, are of size 6×6 and 3×3, and they consist of only 9 and 4 nonzero coefficients. The variation of the coefficients with crack orientation is presented. Equations governing the free transverse and torsion vibrations are derived and solved in both the cases. The formulation has been checked by comparing the theoretical frequencies with the finite element results for a few crack orientations, locations and depths. The agreement is good. It is shown further that, when such cases are analysed for studying the transverse vibration only in one plane by invoking a single rotational spring at the crack location, the approach leads to an erroneous variation of the frequencies with the crack orientations. The data presented here will be useful to solve both forward and inverse problems.