Continuous crack modeling in piezoelectrically driven vibrations of an Euler–Bernoulli beam

One of the main objectives of the structural health monitoring community is to continuously monitor the integrity of a structure by embedding sensors and actuators into the structure. A piezoelectrically driven beam is considered in the present work and the effect of the growth of a single crack on the vibratory characteristics of the beam is studied. It is shown that in order to combine the effect of a crack and the piezoceramic (PZT) actuator into the modeling, available formulations of the mode shapes of the cracked beam, such as modeling the beam as a bi-section connected with a rotational spring, are not applicable as they are not twice differentiable. A new formulation of the mode shapes of the cracked beam is introduced here by applying the Rayleigh–Ritz method. The effect of location and depth of the crack on the vibratory response of the beam is modeled through formulating the loss of energy due to the presence of the crack, which is approximated here as a massless rotational spring. It is shown that the proposed crack modeling provides a close approximation of the mode shapes of the cracked beam. This modeling approach is beneficial as it provides twice-differentiable mode shapes for the cracked beam and can be used as trial functions in the assumed mode approximations. Numerical analysis is also provided for various design parameters, such as crack depth and position and the location of the PZT patch.