On the Dynamics of a Cracked Beam

The dynamic behaviour of a cracked beam is investigated by using the dynamic stiffness method through the application of compliance properties of the crack. The cracked beam is essentially idealised by connecting two Bernoulli-Euler beams at the crack location where a local flexibility matrix is introduced. The dynamic stiffness matrix of the combined system is then developed. The formulation leads to a non-linear eigenvalue problem for which the Wittrick-Williams algorithm is used as solution technique to yield natural frequencies and mode shapes of the cracked beam. The theory developed is general, but for presentational purposes, results for a cantilever cracked beam are illustrated, emphasizing the effects of crack location and crack depth on the natural frequencies and mode shapes. Comparison of results with published literature show good agreement, confirming the validity and accuracy of the method. The theory developed can be extended to frame works and other structures.