Free Vibration Analysis of a Uniform Continuous Beam with an Arbitrary Number of Cracks and Spring-Mass Systems

Abstract

This paper presents a method on free vibration analysis of a uniform continuous beam with an arbitrary number of cracks and spring-mass systems and a damage identification algorithm. Firstly, based on an equivalent model of a single-span beam obtained by using fictitious cracks or fictitious spring-mass systems, mode shape functions of a single-span uniform beam are derived according to fundamental solutions and recurrence formulas. Next, the motion equations and compatibility conditions of spring-mass system at intermediate supports are used to establish the characteristic equation of continuous beam with an arbitrary number of cracks and spring-mass systems. The order of determinant for the characteristic equation is lower, and that of m-spans continuous beam with an arbitrary number of cracks and T spring-mass systems is only $$3m+T-1$$ . Then, according to the formed free vibration analysis method, a damage identification algorithm for the uniform continuous beam with an arbitrary number of cracks and spring-mass systems is presented. This algorithm only utilizes the first-order mode to identify crack position and depth. Finally, the results of this paper are compared with those of the finite element method and other existing references to validate the proposed method. The effects of parameters for cracks and spring-mass systems on natural frequencies of continuous beams are studied. Inverse problem examples of a three-span cracked continuous beam carrying one spring-mass system are also discussed.