Nonlinear Dynamic and Stability Analysis of an Edge Cracked Rotating Flexible Structure

Abstract

The Homotopy Perturbation Method (HPM) is applied to investigate the nonlinear free and forced vibration behavior of the rotating cracked beam. Nonlinear governing equations considering the two-dimensional (2D) large flexible structure in a rotating reference frame considering centrifugal forces are obtained by the Lagrangian approach and the Assumed Mode Method (AMM). The crack is modeled as an elastic nonlinear massless rotational spring, which divides the beam into two parts. The Rayleigh–Ritz method is used to discretize the governing system of equations of the motion. Stability analysis along with bifurcation and phase portrait represents the different behavior of the system, depending on the variations of base angular velocity, crack location, and stiffness. Moreover, it is shown that as the rotational speed increases, a tensile force appears along the neutral axis, stiffening the cracked structure, which results in shifting the backbone to the right and highly affects the nonlinear features of the system. The results obtained through a comparative study of the HPM with first-order approximation and numerical simulations (Runge–Kutta algorithm) demonstrate an accurate and effective solution for structures with nonlinear dynamics.