Generic frequency equation for one-dimensional waveguides with incipient cracks based on the phase closure principle

Abstract

This paper presents a generic frequency equation and its approximate solution for a one-dimensional (1-D) waveguide with incipient open cracks. Extending the principle of phase closure for an intact 1-D waveguide to that for a cracked waveguide, the proposed approach is applicable to any type of cracked 1-D waveguide with arbitrary support conditions. The closed-form frequency equation is simply expressed in terms of phase shifts at the incipient cracks and at both ends of the waveguide, which is physically intuitive and easy to understand from a wave perspective. The validity and implication of the proposed frequency equation is demonstrated for the case of Timoshenko beam with multiple incipient cracks through comparisons with the numerical results from the Timoshenko 1-D model and finite element analysis and with the experimental data.