Condition monitoring of impact vibrations using observed signals and estimated principal Lyapunov exponents

An efficient algorithm for the condition monitoring of a system experiencing impact vibrations is proposed. The condition monitoring of the system is achieved using only observed signals. Delay embedding techniques are used to construct an equivalent system. The Jacobian matrix is approximated using different neighbourhood matrices, and principal Lyapunov exponents are calculated using the QR decomposition method (where Q represents an orthogonal matrix and R represents an upper triangular matrix). The proposed algorithm is then validated using impact velocity series and displacement data. Numerical bifurcation diagrams and the corresponding principal Lyapunov exponents are plotted with respect to various control parameters. The results indicate that sudden changes in impact vibrations can be detected successfully. The principal Lyapunov exponents align with the bifurcation diagram across the entire region. To assess the robustness of the proposed algorithm, the numerical bifurcation diagram is masked by transient noise or noise with a signal-to-noise ratio of 20 dB. Furthermore, the results indicate that the primary regions of stable and unstable impact vibrations can still be detected successfully.