Parametric instability of functionally graded beams with an open edge crack under axial pulsating excitation

This paper studies the parametric instability of functionally graded beams with an open edge crack subjected to an axial pulsating excitation which is a combination of a static compressive force and a harmonic excitation force. It is assumed that the materials properties follow an exponential variation through the thickness direction. Theoretical formulations are based on Timoshenko beam theory and linear rotational spring model. The governing equations of motion are derived by using Hamilton¿s principle and transformed into a set of Mathieu equations through Galerkin¿s procedure. The natural frequencies with different end supports are obtained. The boundary points on the unstable regions are determined by using Bolotin¿s method. Numerical results are presented to highlight the influences of crack location, crack depth, material property gradient, beam slenderness ratio, compressive load, and boundary conditions on both the free vibration and parametric instability behaviors of the cracked functionally graded beams.