Analysis of nonlinear frequency mixing in 1D waveguides with a breathing crack using the spectral finite element method

A breathing crack, due to its bilinear stiffness characteristics, modifies the frequency spectrum of a propagating dual-frequency elastic wave, and gives rise to sidebands around the probing frequency. This paper presents an analytical-numerical method to investigate such nonlinear frequency mixing resulting from the modulation effects induced by a breathing crack in 1D waveguides, such as axial rods and the Euler-Bernoulli beams. A transverse edge-crack is assumed to be present in both the waveguides, and the local flexibility caused by the crack is modeled using an equivalent spring approach. A simultaneous treatment of both the waveguides, in the framework of the Fourier transform based spectral finite element method, is presented for analyzing their response to a dual frequency excitation applied in the form of a tone-burst signal. The intermittent contact between the crack surfaces is accounted for by introducing bilinear contact forces acting at the nodes of the damage spectral element. Subsequently, an iterative approach is outlined for solving the resulting system of nonlinear simultaneous equations. Applicability of the proposed method is demonstrated by considering several test cases. The existence of sidebands and the higher order harmonics is confirmed in the frequency domain response of both the waveguides under investigation. A qualitative comparison with the previous experimental observations accentuates the utility of the proposed solution method. Additionally, the influence of the two constituent frequencies in the dual frequency excitation is assessed by varying the relative strengths of their amplitudes. A brief parametric study is performed for bringing out the effects of the relative crack depth and crack location on the degree of modulation, which is quantified in terms of the modulation parameter. Results of the present investigation can find their potential use in providing an analytical-numerical support to the studies geared towards the inverse analyses based on nonlinear wave-damage interactions.