Firas A. Khasawneh

Increased Stability of Low-Speed Turning Through a Distributed Force and Continuous Delay Model

machining. In the past, this improved stability has been attributed to the energy dissipated by the interference between the workpiece and the tool relief face. In this study, an alternative physical explanation is described. In contrast to the conventional approach, which uses a point force acting at the tool tip, the cutting forces are distributed over the tool-chip interface. This approximation results in a second-order delayed integrodifferential equation for the system that involves a short and a discrete delay. A method for determining the stability of the system for an exponential shape function is described, and temporal finite element analysis is used to chart the stability regions. Comparisons are then made between the stability charts of the point force and the distributed force models for continuous and interrupted turning. DOI: 10.1115/1.3187153

Stability of delay integro-differential equations using a spectral element method

Abstract This paper describes a spectral element approach for studying the stability of delay integro-differential equations (DIDEs). In contrast to delay differential equations (DDEs) with discrete delays that act point-wise, the delays in DIDEs are distributed over a period of time through an integral term. Although both types of delays lead to an infinite dimensional state-space, the analysis of DDEs with distributed delays is far more involved. Nevertheless, the approach that we describe here is applicable to both autonomous and non-autonomous DIDEs with smooth bounded kernel functions. We also describe the stability analysis of DIDEs with special kernels (gamma-type kernel functions) via converting the DIDE into a higher order DDE with only discrete delays. This case of DIDEs is of practical importance, e.g., in modeling wheel shimmy phenomenon. A set of case studies are then provided to show the effectiveness of the proposed approach.

Comparison between collocation methods and spectral element approach for the stability of periodic delay systems

This paper compares two methods that are commonly used to study the stability of delay systems. The first is a collocation technique while the second is a spectral element approach which uses the weighted residual method. Two distributions of the collocation points are compared: the first uses the extrema of Chebyshev polynomials of the first kind whereas the second uses the Legendre-Gauss-Lobatto points. The spectral element approach uses the Legendre-Gauss-Lobatto points and higher-order trial functions to discretize the delay equations while Gauss quadrature rules are used to evaluate the resulting weighted residual integrals. Two case studies are used to compare the different methods. The first case study is a 3rd order autonomous DDE while the second is a DDE describing the midspan deflections of an unbalanced rotating shaft with feedback gain (nonautonomous DDE). Convergence plots that compare the different rates of convergence of the described methods are also provided.

Stability Determination in Turning Using Persistent Homology and Time Series Analysis

This paper describes a new approach for ascertaining the stability of autonomous stochastic delay equations in their parameter space by examining their time series using topological data analysis. We use a nonlinear model that describes the tool oscillations due to self-excited vibrations in turning. The time series is generated using Euler-Maruyama method and then is turned into a point cloud in a high dimensional Euclidean space using the delay embedding. The point cloud can then be analyzed using persistent homology. Specifically, in the deterministic case, the system has a stable fixed point while the loss of stability is associated with Hopf bifurcation whereby a limit cycle branches from the fixed point. Since periodicity in the signal translates into circularity in the point cloud, the persistence diagram associated to the periodic time series will have a high persistence point. This can be used to determine a threshold criteria that can automatically classify the system behavior based on its time series. The results of this study show that the described approach can be used for analyzing datasets of delay dynamical systems generated both from numerical simulation and experimental data.Copyright

Investigation of Period-Doubling Islands in Milling With Simultaneously Engaged Helical Flutes

This paper investigates the stability of a milling process with simultaneously engaged flutes using the state-space TFEA and Chebyshev collocation methods. In contrast to prior works, multiple flute engagement due to both the high depth of cut and high stepover distance are considered. A particular outcome of this study is the demonstration of a different stability behavior in comparison to prior works. To elaborate, period-doubling regions are shown to appear at relatively high radial immersions when multiple flutes with either a zero or nonzero helix angle are simultaneously cutting. We also demonstrate stability differences that arise due to the parity in the number of flutes, especially at full radial immersion. In addition, we study other features induced by helical tools such as the waviness of the Hopf lobes, the sensitivity of the period-doubling islands to the radial immersion, as along with the orientation of the islands with respect to the Hopf lobes. [DOI: 10.1115/1.4005022]

Exploring equilibria in stochastic delay differential equations using persistent homology

This paper explores the possibility of using techniques from topological data analysis for studying datasets generated from dynamical systems described by stochastic delay equations. The dataset is generated using Euler-Maryuama simulation for two first order systems with stochastic parameters drawn from a normal distribution. The first system contains additive noise whereas the second one contains parametric or multiplicative noise. Using Taken’s embedding, the dataset is converted into a point cloud in a high-dimensional space. Persistent homology is then employed to analyze the structure of the point cloud in order to study equilibria and periodic solutions of the underlying system. Our results show that the persistent homology successfully differentiates between different types of equilibria. Therefore, we believe this approach will prove useful for automatic data analysis of vibration measurements. For example, our approach can be used in machining processes for chatter detection and prevention.Copyright

Utilizing Topological Data Analysis for Studying Signals of Time-Delay Systems

This chapter describes a new approach for studying the stability of stochastic delay equations by investigating their time series using topological data analysis (TDA). The approach is illustrated utilizing two stochastic delay equations. The first model equation is the stochastic version of Hayes equation—a scalar autonomous delay equation—where the noise is an additive term. The second model equation is the stochastic version of Mathieu’s equation—a time-periodic delay equation. In the latter, noise is added via a multiplicative term in the time-periodic coefficient. The time series is generated using Euler–Maruyama method and a corresponding point cloud is obtained using the Takens’ embedding. The point cloud is then analyzed using a tool from TDA known as persistent homology. The results of this study show that the described approach can be used for analyzing datasets of delay dynamical systems that are described using constant as well as time-periodic coefficients. The presented approach can be used for signals generated from both numerical simulation and experiments. It can be used as a tool to study the stability of stochastic delay equations for which there are currently a limited number of analysis tools.

Self-Excited Vibrations in a Delay Oscillator: Application to Milling With Simultaneously Engaged Helical Flutes

This paper investigates the stability of a milling process with simultaneously engaged flutes by extending the state-space temporal finite elements method. In contrast to prior works, multiple flute engagement due to both a high depth of cut and a high stepover distance are considered. A particular outcome of this study is the development of a frame work to determine the stability of periodic, piecewise continuous delay differential equations. Another major outcome is the demonstration of different stability behavior at the loss of stability in comparison to prior results. To elaborate more, period doubling regions are shown to appear at relatively high radial immersions when multiple flutes with either a zero or non-zero helix angle are simultaneously cutting.

A State-Space Temporal Finite Element Approach for Stability Investigations of Delay Equations

This paper describes a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems — systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous.Copyright

Explanation for low-speed stability increases in machining: Application of a continuous delay model

This paper investigates the analysis of delay integro-differential equations to explain the increased stability behavior commonly observed at low cutting speeds in machining processes. In the past, this improved stability has been attributed to the energy dissipation from the interference between the workpiece and the tool relief face. In this study, an alternative physical explanation is described. In contrast to the conventional approach, which uses a point force acting at the tool tip, the cutting forces are distributed over the tool-chip interface. This approximation results in a second order delayed integro-differential equation for the system that involves a short and a discrete delay. A method for determining the stability of the system for an exponential shape function is described, and temporal finite element analysis is used to chart the stability regions. Comparisons are then made between the stability charts that use the conventional point force and those that use the distributed force model for continuous and interrupted turning.Copyright