Piecewise Linear Vibrations of a Cracked Beam

Abstract

We consider the piecewise linear (PWL) vibrations in a cracked Rayleigh beam. The change in local stiffness at the crack site due to a mode-1 crack is introduced through a PWL flexural spring such that the local stiffness is higher in the closed crack configuration than in the open crack configuration. However, without loss of generality, we consider the closed crack configuration to be an intact/pristine beam disregarding the contact micromechanics and relative motion of the cracked surfaces. However, the presented method is applicable even when one considers loss of flexural rigidity in the closed crack configuration. Such a model results in slope discontinuity at the crack site in both open and closed crack configurations. It is recognized that the dynamics in these two mutually exclusive configurations are individually linear and support self-adjoint eigenvalue problems. However, the beam experiences the PWL character of the local stiffness at the crack site when it transits from one configuration to another. With this premise, a semi-analytical approach is evolved by invoking the expansion theorem in each of these configurations in terms of their respective orthonormal eigenfunctions. As the beam transits between the configurations governed by a switching condition, the displacement and velocity of the beam are matched at the very instant. The present study is unique in its semi-analytical approach based on the first principles, physical reasoning, mathematical validity, and the generality that it provides for further investigation. We present interesting results emerging in the free vibrations exhibiting energy exchange between nonclosely spaced modes. However, the forced vibrations exhibit resonance close to the ith PWL frequency, defined in terms of the ith eigenfrequencies of both configurations. Finally, a method based on the canonical action–angle (A–A) variables and the method of averaging is devised to study the forced vibrations of the cracked beam by deriving an averaged slow-flow model. We present the comparative results and discuss the limitations of some of these approaches in the study of such dynamical systems.