Eigenvalue Sensitivity of Cracked Beam Structures

Abstract

This paper presents an analytical method for vibration eigenvalue sensitivity of cracked beam structures on the basis of variational principles. A physical model of massless rotational spring is adopted to describe the discontinuity of rotation caused by the crack. The discontinuity is further represented as a distributed loading along the beam in form of singular functions. The matrix characteristic equation and eigenvalue sensitivity equation are derived based on variational principles along with the orthogonal series for beams of arbitrary boundary conditions. Analogous analysis is performed on the axial vibrations of a beam. Meanwhile, an exact analytical solution of eigenparameters for a cracked simply supported beam is provided to qualitatively study the effect of crack height on the accuracy of the sensitivity equation. The significance of the proposed method is that it can be easily extended to the case where the beam has multiple cracks by superposition of loads. A simply supported beam is analysed by the proposed method and the exact method. The comparison of results is satisfactory and the factors affecting the accuracy are discussed.