Damping parameter estimation using topological signal processing

Abstract

Energy is dissipated in engineering systems through a variety of different, typically complex, damping mechanisms. While there are several common damping models with physical interpretations, it is often necessary to examine noisy, experimental data in order to identify and fit the damping parameters. One class of methods fits the damping parameters based on the decay envelope of the signal’s peaks based on some assumed damping mechanism. While there exist results in the literature, especially for viscous damping, to guide the selection of the spacing between the peaks for the optimum damping ratio identification, these methods often overlook the difficult and classic problem of identifying the signal’s “true” peaks in the presence of noise. Therefore, in this work we utilize tools from Topological Data Analysis (TDA) to address the problem of robust and automatic power-law damping identification from decay or free-response function. The decay function can be obtained from initial excitation, or using random decrement method thus making our approach a viable tool for operational modal analysis. We present our approach using one-dimensional examples, and describe extensions to multiple degree of freedom systems. Our damping identification framework uses the persistent homology of zero dimensional (0D) sub-level sets, one of the main tools of TDA, to separate the true topological features of the signal (i.e. the peaks and valleys) from measurement noise. Using persistent homology we are able to automatically represent the damped response in a noise-robust, two-dimensional summary of the peak–valley pairs called the persistence diagram. The persistence diagram is then analyzed using two methods: (1) a theoretical analysis of the significant persistence pairs and (2) function fitting to the persistence space. We present theoretical results that establish the class of general power-law damping terms where our approach applies, and develop specific tools to identify damping parameters for viscous, coulomb, and quadratic damping. We show that our approach is computationally fast, operates on the raw signal itself without requiring any pre-processing, and reduces the number of decisions needed by the user to perform the needed calculations. The results are validated using a combination of numerical and experimental data.