Natural frequencies of vibration in cracked Timoshenko beams within an elastic medium

Abstract

This work analyzes the dynamic behavior of cracked Timoshenko beams immersed in a Winkler elastic medium, obtaining their natural frequencies of bending vibration. For this purpose, the beam is modeled as two segments connected in the cracked section by two massless springs, one extensional and another one rotational. Their stiffnesses are proportional to the shear force and bending moment transmitted through the cracked section, respectively, considering the discontinuity of vertical displacement and rotation due to bending. The differential equations for the free vibration are solved by applying the corresponding boundary conditions and the compatibility conditions of the cracked section. The proposed methodology allows calculating the natural frequencies of vibration as a function of the type of support, the parameter defining the stiffness of the elastic soil, the crack position and the initial length of the crack. The results obtained are compared with those published by other authors who model the crack in a simplified way, showing the interaction and importance of the parameters considered in the natural frequencies of the system.