Attractor reconstruction of an impact oscillator for parameter identification

This paper presents a simple and novel approach, based on nonlinear time series analysis of an experimental system, to infer from subtle alteration of the system dynamics the changes caused in the system parameters. Using the acceleration time-series as a measurement of simulated and experimental impact oscillators (serving as a model for the drilling conditions with intermittent contact between the drill bit and the formation), the systems attractor is reconstructed and characterised. It is shown that the stiffness correlates with the topology of the reconstructed attractor. Non-impacting trajectories form an approximate plane within the three dimensional reconstructed phase-space, and contact with the constraint causes a systematic deviation from the linear subspace, the inclination of which, measured by the statistics of the tangent vector, can be used to infer the stiffness. This relationship between the topology of attractor and the stiffness was also verified experimentally. Based on the developed framework, it is now possible to classify the stiffness of the impacted material from a single variable in a simple way and in real time.