Parametric Identification of Delay Systems from Empirical Stability Information

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The work presented in this document alleviates challenges associated with applying academically proven analysis techniques to real-world dynamical systems. Specifically, by providing improved parameter identification methods, the contributions of this work help bridge the gap between theoretical predictions and experimentally-observed behavior. This work is primarily motivated by the existence of powerful predictive methods in the literature of manufacturing research, specifically pertaining to the stability of subtractive manufacturing methods like milling and turning, which are not currently implemented in industry due to the prohibitive effort required by available methods for identifying system properties covering the range of cutting conditions that may be encountered. The motivating technique, temporal finite element analysis (TFEA), is first detailed and demonstrated to be sensitive to system parameters. The remainder of this document, subsequently, focuses on improving or developing approaches for identifying parameters influencing or quantifying the level of stability of dynamical systems, offering a connection between theory and experiment.

First, an improvement to the fundamental technique of logarithmic decrement damping estimation is provided. Specifically, an analytical expression for the optimal number of periods between samples is derived. This expression, obtained from an uncertainty analysis of the method's principal equation, is shown to be a function of only one system parameter: the damping ratio. This suggests that for linear systems with viscous damping there is a unique, damping-dependent period choice corresponding to minimum uncertainty in the estimated damping ratio. This result led to the discovery of a constant optimal amplitude ratio offering a straightforward guideline that can be applied by the experimenter to obtain damping ratio estimates with minimum uncertainty. The derived expressions are applied to a series of numerical and experimental systems to confirm their validity.

Next, focus is placed on systems with periodic steady-state behavior. This work applies empirical Floquet theory to extract characteristic multipliers from time series data and builds upon prior works by significantly reducing the influence of experimental noise. Characteristic multipliers (CMs), which are the eigenvalues of the transition matrix governing the evolution of transient solutions over one period of motion, quantify the local stability of periodic orbits and can be estimated experimentally without knowledge of the system model. Traditionally, empirical CM estimates were obtained from the transient dynamics observed by subtracting measured steady-state behavior from a recorded perturbed response. The subtraction of two experimentally measured quantities amplifies noise, producing inaccurate CM estimates. By applying a moving integral to isolate the transient dynamics, this work provides an approach to reduce, rather than amplify, the influence of noise on empirical CM results. This approach is applied to the reconstructed phase space of a numerical example and an experimental system to demonstrate the improvements.

This work culminates in a study identifying parameters of milling operations. In this work, a heuristic optimization routine is developed which, given information gleaned from vibration data collected during cutting, identifies modal parameters and cutting coefficients of the model governing interrupted cutting. Characteristic multipliers estimated from the transient induced by the onset of cutting forces provide information regarding the true stability characteristics of the system. Model parameters are fit to the collected empirical CMs with a genetic algorithm, which is well-suited to navigate the numerous local minima of the objective function. New insight is provided into how control parameters and modal properties affect stability, demonstrating that results can be influenced through careful selection of cutting tests. The approach is applied to single-degree-of-freedom and two-degree-of-freedom milling systems, where accurate estimates of parameters are achieved.





Little, Jared Andrew (2019). Parametric Identification of Delay Systems from Empirical Stability Information. Dissertation, Duke University. Retrieved from


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