Characterization and detection of parameter variations of nonlinear mechanical systems

Abstract

The present study applies the recently developed ideas in experimental system modeling to both characterize the behavior of simple mechanical systems and detect variations in their parameters. First, an experimental chaotic time series was simulated from the solution of the differential equation of motion of a mechanical system with clearance. From the scalar time series, a strange attractor was reconstructed optimally by the method of delays. Optimal reconstructions of the attractors can be achieved by simultaneously determining the minimal necessary embedding dimension and the proper delay time. Periodic saddle orbits were extracted from the chaotic orbit and their eigenvalues were calculated. The eigenvalues associated with the saddle orbits are used to estimate the Lyapunov exponents for the steady state motion. An analysis of the associated one dimensional delay map, obtained from the chaotic time series, is made to determine the allowable periodic orbits and to yield an estimate of the topological entropy for the positive Lyapunov exponent. Sensitivity of the positions of the low order unstable periodic orbits (orbits of short period) of a chaotic attractor is used as a basis for detection of parameter variations in another unsymmetric bilinear system. For the experimental scalar time series generated by the dynamical system as a parameter varies, the chaotic attractors were again optimally reconstructed using the method of delays. The parameter variations were detected by the changes in location of the unstable periodic orbits extracted from the reconstructed attractors of the experimental scalar time series.