Geometrically non-linear free vibrations of clamped–clamped beams with an edge crack

It is well known that a crack in a beam induces a drop in its natural frequencies and affects its modes shapes. This paper is a theoretical investigation of the geometrically non-linear free vibrations of a clamped–clamped beam containing an open crack. The approach uses a semi-analytical model based on an extension of the Rayleigh–Ritz method to non-linear vibrations, which is mainly influenced by the choice of the admissible functions. The general formulation is established using new admissible functions, called “cracked beam functions”, and denoted as “CBF”, which satisfy the natural and geometrical end conditions, as well as the inner boundary conditions at the crack location. Iterative solution of a set of non-linear algebraic equations is obtained numerically, which leads to the basic function contribution coefficients to the displacement response function. Then, an explicit solution is derived and proposed as an alternative procedure, simple and ready to use for engineering applications. Emphasis is made on the backbone curves, i.e. amplitude-frequency dependence, obtained for various crack depth, and the effect of the vibration amplitudes upon the non-linear mode shapes of a cracked beam is examined. The work is restricted to the fundamental mode in order to concentrate on the study of the influence of the crack on the non-linear dynamic response near to the fundamental resonance.