Vibratory characteristics of multi-step beams with an arbitrary number of cracks and concentrated masses

Abstract

A new exact approach for free vibration analysis of a multi-step beam with an arbitrary number of cracks and concentrated masses is proposed. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in the beam. Based on the rotational spring modeling, the fundamental solutions and recurrence formulas developed in this paper, the mode shape function of vibration of a multi-step beam, each step beam has an arbitrary number of cracks and concentrated masses, can be easily determined. The main advantage of the proposed method is that the eigenvalue equation of a multi-step beam with any kind of two end supports, any finite number of cracks and concentrated masses can be conveniently determined from a second order determinant. As a consequence, due to the decrease in the determinant order as compared with previously developed procedures, the computational time required by the present method for solving the title problem can be reduced significantly. A numerical example is given to illustrate the application of proposed methods and to investigate the effect of cracks on natural frequencies and mode shapes of a multi-step beam.