Repair of Free Vibration Behavior of a Cracked Rotating Timoshenko Beam Using a Piezoelectric Patch and Applying Differential Transform Method

This paper presents an analytical method for the application of piezoelectric patches to repair a rotating cracked beam. The beam equations of motion are obtained based on the Timoshenko beam theory including the effects of shear deformation and rotary inertia. The criterion applied for the repair is to modify the first natural frequency of the cracked beam towards that of the healthy beam applying a piezoelectric patch. Due to this, an external voltage is applied to actuate a piezoelectric patch bonded on the beam that decreases the effect of the crack on the vibration characteristics of the beam. First, the coupled equations of motion are discretized by applying the assumed modes method. Then, the cracked beam is modeled as numbers of healthy segments connected by two linear springs at the crack locations (one, extensional and the other, rotational). The compatibility requirements on the crack section and on the ends of the piezoelectric patch are considered to obtain the relationships between any two spans. Finally, applying the semi-analytical differential transform method, the natural frequencies and mode shapes of the system can be calculated. Numerical simulations are performed to assess the effects of different conditions on the repair moment coefficient. The presented model is validated by comparing the results with those available in the literature where, the natural frequencies are in a reasonably good agreement with the reported results.