Stability Analysis of a Cracked Blade Coupled with a Rigid Rotor

Flexible blades coupled to rotating systems are commonly used in industrial machines, such as compressors, exhausters, and turbines. These components are usually exposed to different operating conditions, including high speed, large centrifugal forces, high temperatures, and pressure. Considering the inevitable manufacturing flaws, cracks can emerge and grow particularly in blades of these systems. Thus, investigations on the dynamic behavior of cracked blades become mandatory to prevent failures. In this work, the development, solution, and instability analysis of a system composed of four flexible blades coupled to a flexible shaft are presented. The flexible blades are modeled as Euler-Bernoulli beams with tip masses attached at their ends. Their deformations are obtained by considering second order nonlinear terms to ensure that the centrifugal stiffness is correctly represented, thus characterizing a second order linearized model. The equations of motion are obtained by applying the so-called Newton-Euler-Jourdain method. The crack presence brings an additional flexibility to the blades, which is introduced in the model by using a torsional spring. The resulting blade stiffness is obtained through the beam elastic equation. The Newmark time integration method is associated with the Newton-Raphson iteration procedure to integrate the equations of motion. The system was evaluated for different situations, regarding the depth of the crack in the blades, as well as the operating condition of the rotor-blade system. Finally, the instability map and the vibration responses of the system is determined. The obtained results indicate the instability condition of the rotor-blade system for a certain combination of rotating speed, angular position of the blades, and crack depth.