Variational formulation of a rod under torsional vibration for crack identification

A variational formulation is developed for the torsional vibration of a cylindrical shaft with a circumferential crack. The work is compared with existing methods. The Hu-Washizu-Barr variational formulation was used to develop the differential equation and the boundary conditions of the cracked rod. The general variational principle and the independent assumptions about the displacement, the momentum, the strain and the stress fields of the cracked rod, and the equations of motion for a uniform rod in torsional vibration, are derived. The crack was modelled as a continuous flexibility using the displacement field in the vicinity of the crack, found with fracture mechanics methods. Rayleigh quotient was used to approximate the natural frequencies of the cracked rod. Independent evaluations of crack identification methods in rotating shafts are reported and compared with methods using the continuous crack flexibility theory.