Vibration, Buckling and Dynamic Stability of Tapered Cantilever Beam with Single Open Transverse Crack

Abstract

Present research mainly deals with the vibration, buckling and parametric instability behavior of a tapered cantilever beam subjected to single open transverse crack. For the sake of relevance and completeness of present study parametric instability behavior is also done for other support conditions. Variation of free vibration frequencies, buckling load and dynamic stability regions with respect to depth, position of the crack, static and dynamic load factors are investigated utilizing finite element method (FEM) through Bolotin's methodology. A simple harmonically fluctuating in-plane axial load is considered to be acting on the beam. The equation of motion during instability represents a second order differential condition with periodic coefficient of the Mathieu-Hill type. Floquet's hypothesis is utilized for the development of instability region and periodic solutions are obtained by using finite element method according to Bolotin's approach. Derivation of stiffness matrix of cracked beam element is done in line with Zheng (2004) where the derivation of stiffness matrix of cracked uniform beam is described. Presently, for the tapered cracked beam element, stiffness matrix is obtained from the total flexibility matrix of cracked beam element. A MATLAB code is developed utilizing finite element method to obtain the natural frequencies, buckling loads and excitation frequencies of the tapered cracked beam. It is observed that the free vibration frequencies for tapered cantilever beam with single open transverse crack for different modes of vibration depends on position of crack from fixed end and relative depths of crack. Similarly, in buckling analysis buckling load variation is almost same as that of frequency. In dynamic stability analysis of tapered beam with single open transverse crack, onset of dynamic instability and widening of dynamic instability regions are studied for different parameters. It was found that various parameters affect the dynamic stability of beam in different ways and two boundaries of excitation frequencies is obtained, it implies that any structural component will fail if their frequency lie within that boundary region before attaining the resonance frequency. Thus for the safety, reliability and operational existence of structures the dynamic stability behavior of crack structural components is very important.