This paper addresses the study of the nonlinear dynamics of non-smooth systems representative of beams with breathing cracks. The aim is to use the nonlinear characteristics of the system response to identify the damage in cracked structures that behave similarly to bilinear systems and hence exhibit nonlinear phenomena in the dynamic response even for low damage levels. The idea is supported by the study of a piecewise smooth 2-DOF model where a wide variety of nonlinear phenomena has been evidenced, which include among others the bifurcations of super-abundant modes and a number of resonances greater than the system degrees of freedom. All these phenomena are strongly dependent on the stiffness discontinuity which is governed by the damage parameter. A novel method able to detect crack severity and position through measurements of the system nonlinear response has been developed and a cantilever beam with a breathing crack is considered as a numerical test case. The inverse procedure is tested by identifying the position and depth of a crack using pseudo-experimental data; the results show a strong robustness of the method even in the case when the data are affected by measurement errors.
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