On the Spectral Signature of Weakly Bilinear Oscillators

Abstract

An analytical relationship is derived between the strength of a weak stiffness discontinuity and the magnitudes of superharmonic peaks in the Fourier spectrum of a bilinear oscillator. Expressions are obtained for three ranges of forcing frequencies: far from resonances, and near primary and superharmonic resonances. The difference between the two linear stiffness coefficients is used as the small parameter, ε, in a modified perturbation analysis. Closed-form expressions for the extra harmonic components in the spectrum are obtained as functions of the strength of the stiffness discontinuity. These results are relevant to the problem of detecting small cracks in structures and shafts; they suggest that greater sensitivity to small cracks can be obtained by exploiting superharmonic resonance.