Generalized multi-symplectic method for vibration of cracked simply supported beam

The effects of cracks on the dynamic behavior of engineering structures affect the structure safety. In this paper, a generalized multi-symplectic approach is developed to investigate the effects of crack on a dynamic behavior of a simply supported beam in detail. First, the effect of the crack on the bending stiffness of the simply supported beam is expressed by an equivalent bending stiffness and the dynamic model with variable coefficients is established to describe the transverse vibration of the cracked beam. Then, the approximate symmetric form of the dynamic model is constructed and its dynamic symmetry breaking factors are discussed. The Preissmann scheme of the approximate symmetric form is constructed based on the generalized multi-symplectic theory. In numerical simulations, the structure-preserving properties of the Preissmann scheme are illustrated and the validity of the dynamic model is verified. Then, the effects of the crack on the vibration of the beam are investigated by using the generalized multi-symplectic method in detail. The numerical results presented prove that the proposed generalized multi-symplectic method is an effective approach to investigate the dynamic behavior of cracked structures.