Dynamic Stability Analysis of a Cracked Beam using a Lumped Parameter Model

Abstract

Vibration based condition monitoring is one of the predictive maintenance strategies applied for structures and rotating machinery. Detection and identification of fatigue crack in structures has been a widely discussed problem in the literature of condition monitoring. This paper discusses the numerical results that have been obtained from the study of bending vibration of a cracked beam. The crack is primarily modelled as an opening/ closing crack resulting in a periodic variation in stiffness. The mathematical model considered for the cracked beam is a lumped single degree of freedom system. The resulting system equation of motion has the canonical form of a Mathieu equation. For obtaining the system response, the equation is written in a state-space form and then solved using fourth order Runge-Kutta numerical integration. The time and frequency domain responses are obtained for a specific harmonic excitation. Further, the stability of the cracked beam is discussed with respect to the variation in stiffness, by numerical analysis, using a Vander Pol and Strutt stability chart diagram.