On the dynamics of a beam with switching crack and damaged boundaries

The local equivalent linear stiffness method (LELSM), which has previously been found to be efficient in predicting the modal behavior of nonlinear structural dynamic systems with cubic and dead-zone nonlinearities, is applied to a beam with a switching crack where the crack is modeled as a nonsmooth (clearance) nonlinearity. In addition, the LELSM is also applied for the boundary damage of the beam where the damage is modeled via smooth (cubic) nonlinearity, while a new technique for finding the global equivalent bilinear stiffness matrix (GEBSM) is introduced here to be applied to the systems with clearance nonlinearity or switching stiffness. As a result, the GEBSM of the system of concern is formulated based on the approximated analytical values of the bilinear frequencies of the system. For a system with mixed clearance and cubic nonlinearities, the GEBSM is found first for the clearance nonlinearity, and then the LELSM is subsequently applied to update the GEBSM for finding the global equivalent linear stiffness matrix (GELSM) of the system. It is found that the GEBSM and GELSM modes, for which no a priori simulation is required, closely approximate the corresponding proper orthogonal modes for either a vanishing clearance or a small nonvanishing clearance.