Nonlinear Vibrations of a Beam with a Breathing Edge Crack Using Multiple Trial Functions

Abstract

In this paper, a beam like structure with a single edge crack is modeled and analyzed in order to study the nonlinear effects of breathing crack on transverse vibrations of a beam. In literature, edge cracks are generally modeled as open cracks, in which the beam is separated into two pieces at the crack location and these pieces are connected to each other with a rotational spring to represent the effect of crack. The open edge crack model is a widely used assumption; however, it does not consider the nonlinear behavior due to opening and closing of the crack region. In this paper, partial differential equation of motion obtained by Euler-Bernoulli beam theory is converted into nonlinear ordinary differential equations by using Galerkin’s method with multiple trial functions. The nonlinear behavior of the crack region is represented as a bilinear stiffness matrix. The nonlinear ordinary differential equations are converted into a set of nonlinear algebraic equations by using harmonic balance method (HBM) with multi harmonics. Under the action of a harmonic forcing, the effect of crack parameters on the vibrational behavior of the cracked beam is studied.