Random Vibrations of a Cracked Beam Based on a Discrete Spring Model

Abstract

Considering the crack as a massless rotational spring, the equivalent stiffness of the beam with an arbitrary number of cracks is described by the generalized function. The explicit analytical expressions of the mode functions of the transverse displacement and bending moment for the cracked beam are presented. The methods to calculate the power spectral density and variance of deflection and moment for the cracked beam subjected to a random excitation are studied. The numerical results show that the results calculated in this paper are in good agreement with those of Monte-Carlo simulation. Meantime, with the crack depth increasing, the curve peaks of deflection power spectral density and bending moment power spectral density of the cracked beam increase, the corresponding frequencies reduce, and the variances of mid-span deflection and bending moment increase as well.