Nonlinear forced vibration analysis of a multi-cracked Euler-Bernoulli curved beam with inclusion of damping

Curved beams that have advantages of excellent bearing capacity and beautiful appearance are widely applied in various engineering constructions. This paper studies nonlinear forced vibration of a multi-cracked Euler-Bernoulli curved beam (ECB) with damping effects and derives the closed-formed analytical solution of steady-state forced vibration of the multi-cracked curved beam by means of Green’s functions. Based on the mathematical structure of Green’s functions, a new transfer matrix method called the opposite coordinate two-step transfer matrix (OCTM) method is proposed. By employing the OCTM method and Laplace transform method, Green’s functions of ECBs with one crack and multiple cracks are obtained. The vibration model of the multi-cracked ECB degenerates into a healthy curved beam vibration model when depths of cracks are set to zero. The healthy curved beam vibration model can be reduced to a straight beam vibration model by setting the radius of curvature to infinity. The present analytical solution is verified by experimental results, finite element results, and analytical solutions in the literature in this work. Effects of some important physical parameters, such as the crack depth and location, on vibration of a curved beam with one crack are investigated. Interaction between two cracks of a curved beam is also investigated.