Recursive formulations for dynamic analysis of a smart cracked beam based on the method of reverberation-ray matrix

An accurate electro-mechanical impedance (EMI) model for analytical crack detection of a Timoshenko beam is established based on the improved reverberation-ray matrix method via the recursive formulations. The crack is treated as a massless rotational spring. A pair of PZT patches with free ends symmetrically bonded onto the beam is assumed in a state of pure one-dimensional axial motion under an out-of-phase harmonic electric force and hence a pure bending excitation of the beam is created. A Kelvin-type viscoelastic law is introduced into the EMI model to account for the bonding effect in order to improve the signature sensitivity to cracks. In order to develop the recursive formulations to reduce the size of the final equations in the method of reverberation-ray matrix (MRRM), new scattering relations are established by a matrix reduction technique. Then, an analytical expression of impedance containing crack parameters as well as other physical parameters is derived for crack detection of beams.