Dynamics of a cantilever beam with solitary open crack: nonlinear response to external and base excitation

We employ a well-established theoretical modeling approach to simulate the dynamics of a cantilever beam with an open crack subjected to simultaneous external and parametric excitation while accounting for the shortening effect. The existing crack modeling formulation involves modifying the mode shape based on the concept of stiffness singularity arising from the crack. Combining this approach with 2D nonlinear beam theory, we derive the equation of motion. Subsequently, the system is discretized using Galerkin projection for modal analysis of both the cracked and regular beams. Our study provides insights into the effects resulting from the complementary behavior of shortening and crack-induced singularity on the modal frequencies, creating conditions favorable for internal and combination resonance. To explore the system’s behavior, we utilize a simple shooting method to identify periodic limit sets. Alongside arc-length-based continuation, this method helps determine the periodic solution manifold for forced response. The validity of the results is confirmed through verification using a brute-force continuation approach. Additionally, the study captures typical 2:1 and 3:1 internal resonance behavior between the second and third modes of the cracked beam. We gain insights into the stability of periodic solutions near resonance zones through stability charts for all cases.