VIBRATION AND STABILITY OF A CRACKED TRANSLATING BEAM

Abstract

The vibration and stability characteristics of a cracked beam translating between fixed supports are investigated. Using Hamilton's principle and elementary fracture mechanics, the equations of motion for the beam are developed. Throughout this analysis it is assumed that the crack is shallow and always remains open, i.e., crack closure and the associated impact conditions are not considered. In order to restrict attention to the open crack scenario, parameter regimes corresponding to (1) a fully open crack, (2) a fully closed crack, and (3) a partly open–partly closed crack are clearly identified. For parameter values in regime (1), the free vibration characteristics are studied via an eigenanalysis. This shows that the natural frequencies (Im (λ)) and stability characteristics (Re (λ)) fluctuate as the crack translates along with the beam between the two supports. For the shallow cracks being considered, the fluctuations are attributed primarily to the localized drop in the mass per unit length (occurring at the crack) rather than from the increased flexibility. Furthermore, the magnitudes of these fluctuations are shown to vary with both the axial transport speed and the crack depth and are mapped in the control parameter space. Implications for the free and forced vibration problems are discussed.