A Dynamical Systems Approach to Damage Evolution Tracking, Part 2: Model-Based Validation and Physical Interpretation

In this paper, the hidden variable damage tracking method developed in Part 1 is analyzed using a physics-based mathematical model of the experimental system: a mechanical oscillator with a nonstationary two-well potential. Numerical experiments conducted using the model are in good agreement with the experimental study presented in Part 1, and explicitly show how the tracking metric is related to the slow hidden variable evolution responsible for drift in the fast system parameters. Using the idea of averaging, the slow flow equation governing the hidden variable evolution is obtained. It is shown that the solution to the slow flow equation corresponds to the hidden variable trajectory obtained with the experimental tracking method. Thus we establish in principle the relationship of our algorithm to any underlying physical process. Based on this result, we discuss the application of the tracking method to systems with evolving material damage using the results of some preliminary experiments.@DOI: 10.1115/1.1456907# In Part 1 of this paper, motivated by the need to track damage evolution in machinery, we have developed a nonlinear method for tracking slowly evolving hidden variables. From this perspective, damage is a hidden process causing nonstationarity in a fast, directly observable dynamical system. The method uses a phase space formulation of the damage tracking problem, and uses a tracking metric developed using the short-time reference model prediction error. The method was successfully applied to an electromechanical experimental system consisting of a vibrating beam with a nonlinear potential perturbed by a battery powered electromagnet. The connection between the tracking metric developed in Part 1 and the hidden drift state variable was demonstrated empirically. It was shown that, as expected from the theoretical derivation of the method, the tracking metric is in a one-to-one relationship with the local time average of the measured voltage signal. In this, Part 2, of our paper, a physics based mathematical model of the experimental system is used to study analytically the direct connection between the tracking metric and the hidden drift process. Numerical experiments performed with the model are used to validate the experimental method. The idea of averaging is then used to show that the output of the tracking method is in fact following the solution to the slow flow equation for the drifting process. This provides a physical interpretation for the output of the tracking algorithm, and shows how, in principle, the experimental method can be related to the physics of the damage process. Based on this physical interpretation, we return to the experimental application of the tracking method, and discuss some preliminary results for a system with a crack growing to failure. In the next section, we develop the mathematical model of the battery discharge experiment using a lumped parameter, Lagrangian formulation of the electromechanical system. In Section 3, the tracking method developed in Part 1 is applied to the mathematical model in numerical experiments. Using the output from numerical simulations of the mathematical model, we are able to validate the tracking algorithm. In Section 4, we discuss the method of averaging as it relates to our problem. Finally, in Section 5, we finish with concluding remarks.