Nonlinear Dynamic Analysis of a Simply Supported Beam with Breathing Crack Using Proper Orthogonal Decomposition Based Reduced-Order Modeling

The nonlinearity associated with a bridge is considered and the bridge is modeled as a simply supported beam (SSB) with breathing crack in Abaqus CAE Environment. The breathing mechanism (opening and closing) of the crack is achieved by the application of periodic loading in the transverse direction, in out of plane orientation to the crack and in parallel to the crack orientation of the beam. In this study, damage initiation (crack propagation) is neglected and the research is carried out only for the static crack. Using FEA, the higher order PDEs are converted into a set of coupled ODEs using Galerkin’s weak formulation. The refined mesh is adopted near the crack tip zone and intentionally increases the computational time. In order to reduce the simulation time without losing its accuracy, reduced-order modeling (ROM) approach is implemented in the cracked model to obtain the computationally efficient and equivalent model for the dynamic analysis. With this, we capture more than 99% of the system energy using subspace projection on to the full domain with two proper orthogonal decomposition (POD) modes. Additionally, the effective properties (mass, linear, and nonlinear stiffness, damping, and forcing amplitude) of the nonlinear dynamical system are obtained and incorporated into the Duffing oscillator’s equation of motion, with a cubical stiffness, which is identified from the static deflection of the beam and is solved using state-space approach using Matlab ode45 Algorithm. However, the parametric study is also conducted for different types of forcing amplitudes and frequencies to check the consistency of the ROM with the FEA Abaqus simulation and is in good agreement with its responses. The aperiodic behavior is identified in the linear range and the periodic doubling route to chaos is identified in the nonlinear range qualitatively in this model.