A Unique Methodology for Chatter Stability Mapping in Simultaneous Machining

2005 ◽  
Vol 127 (4) ◽  
pp. 791-800 ◽  
Author(s):  
Nejat Olgac ◽  
Rifat Sipahi
Keyword(s):  
Ad Hoc ◽  
Metal Removal ◽  
Cutting Tools ◽  
Chip Thickness ◽  
Delay Systems ◽  
Mathematical Framework ◽  
Multiple Time ◽  
Machined Surface ◽  
Characteristic Roots ◽  
The Stability

A novel analytical tool is presented to assess the stability of simultaneous machining (SM) dynamics, which is also known as parallel machining. In SM, multiple cutting tools, which are driven by multiple spindles at different speeds, operate on the same workpiece. Its superior machining efficiency is the main reason for using SM compared with the traditional single tool machining (STM). When SM is optimized in the sense of maximizing the rate of metal removal constrained with the machined surface quality, typical “chatter instability” phenomenon appears. Chatter instability for single tool machining (STM) is broadly studied in the literature. When formulated for SM, however, the problem becomes notoriously more complex. There is practically no literature on the SM chatter, except a few ad hoc and inconclusive reports. This study presents a unique treatment, which declares the complete stability picture of SM chatter within the mathematical framework of multiple time-delay systems (MTDS). What resides at the core of this development is our own paradigm, which is called the cluster treatment of characteristic roots (CTCR). This procedure determines the regions of stability completely in the domain of the spindle speeds for varying chip thickness. The new methodology opens the research to some interesting directions. They, in essence, aim towards duplicating the well-known “stability lobes” concept of STM for simultaneous machining, which is clearly a nontrivial task.

Chatter Stability Mapping for Simultaneous Machining

2005 ◽  
Author(s):  
Nejat Olgac ◽  
Rifat Sipahi
Keyword(s):  
Ad Hoc ◽  
Metal Removal ◽  
Cutting Tools ◽  
Delay Systems ◽  
Work Piece ◽  
Mathematical Framework ◽  
Multiple Time ◽  
Machined Surface ◽  
Characteristic Roots ◽  
The Stability

A novel analytical tool is presented to assess the stability of simultaneous machining (SM) dynamics, also known as parallel machining (PM). In SM, multiple cutting tools, which are driven by multiple spindles at different speeds, operate on the same work-piece. Its superior machining efficiency is the main reason for using SM compared with the traditional single tool machining (STM). When SM is optimized in the sense of maximizing the rate of metal removal constrained with the machined surface quality, typical “chatter instability” phenomenon appears. Chatter instability for single tool machining (STM) is broadly studied in the literature. When formulated for SM, however, the problem becomes notoriously more complex. There is practically no literature on the SM chatter, except a few ad-hoc and inconclusive reports. This study presents a unique treatment, which declares the complete stability picture of SM chatter within the mathematical framework of multiple time-delay systems (MTDS). We utilize our recent methodology, called the Cluster Treatment of Characteristic Roots (CTCR), which is developed for this general class of dynamics. As an end result CTCR offers the regions of stability completely in the domain of the spindle speeds. This new methodology opens the research to some interesting directions. They, in essence, aim towards duplicating the “stability lobes” concept of STM for SM, which is clearly a nontrivial task.

Robust Control for Multiple Time Delay Systems With Delay–Decouplability Concept

2008 ◽  
Author(s):  
Kamran Turkoglu ◽  
Nejat Olgac
Keyword(s):  
Characteristic Equation ◽  
Linear Time ◽  
Robustness Analysis ◽  
Delay Systems ◽  
Multiple Time ◽  
Multiple Control ◽  
Pendulum System ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
The Stability

We consider linear time-invariant minimum phase MIMO plants in this paper, with multiple control delays. The delays appear at several components of the state. Deployment of delay decoupling control (DDC) creates a characteristic equation which facilitates the assessment of stability in each of the delays independently from each other. When, however, some system parameters are uncertain, the characteristic equation seems to entail truly coupled delays, which forces the stability assessment to an N-P hard complexity class problem. We show that this assessment can be very efficient using the Cluster Treatment of Characteristic Roots (CTCR) paradigm. The main contribution of the study is for a certain class of structures, if the feedback control forms with independent delays on separate feedback channels decouplability may still hold, and the robustness analysis becomes efficient. This result is demonstrated for 2-input, 2-output system, and it is claimed that the findings are scalable to higher dimensional dynamics. Example case study of a cart-pendulum system is treated.

Kernel and Offspring Concepts for the Stability Robustness of Multiple Time Delayed Systems (MTDS)

2006 ◽  
Vol 129 (3) ◽  
pp. 245-251 ◽  
Author(s):  
Rifat Sipahi ◽  
Nejat Olgac
Keyword(s):  
Time Delays ◽  
Linear Time ◽  
Characteristic Root ◽  
Invariance Property ◽  
Multiple Time ◽  
Delayed Systems ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
Stability Robustness ◽  
The Stability

A novel treatment for the stability of linear time invariant (LTI) systems with rationally independent multiple time delays is presented in this paper. The independence of delays makes the problem much more challenging compared to systems with commensurate time delays (where the delays have rational relations). We uncover some wonderful features for such systems. For instance, all the imaginary characteristic roots of these systems can be found exhaustively along a set of surfaces in the domain of the delays. They are called the “kernel” surfaces (curves for two-delay cases), and it is proven that the number of the kernel surfaces is manageably small and bounded. All possible time delay combinations, which yield an imaginary characteristic root, lie either on this kernel or its infinitely many “offspring” surfaces. Another hidden feature is that the root tendencies along these surfaces exhibit an invariance property. From these outstanding characteristics an efficient, exact, and exhaustive methodology results for the stability assessment. As an added uniqueness of this method, the systems under consideration do not have to be stable for zero delays. Several example case studies are presented, which are prohibitively difficult, if not impossible to solve using any other peer methodology known to the authors.

Stability Switches of a Class of Fractional-Delay Systems With Delay-Dependent Coefficients

2018 ◽  
Vol 13 (11) ◽  
Author(s):  
Xinghu Teng ◽  
Zaihua Wang
Keyword(s):  
Time Delay ◽  
Delay Systems ◽  
Jacobian Determinant ◽  
Stability Switches ◽  
Characteristic Roots ◽  
Fractional Delay ◽  
Analysis Of Stability ◽  
The Stability ◽  
Key Steps ◽  
Delay Dependent

Stability of a dynamical system may change from stable to unstable or vice versa, with the change of some parameter of the system. This is the phenomenon of stability switches, and it has been investigated intensively in the literature for conventional time-delay systems. This paper studies the stability switches of a class of fractional-delay systems whose coefficients depend on the time delay. Two simple formulas in closed-form have been established for determining the crossing direction of the characteristic roots at a given critical point, which is one of the two key steps in the analysis of stability switches. The formulas are expressed in terms of the Jacobian determinant of two auxiliary real-valued functions that are derived directly from the characteristic function, and thus, can be easily implemented. Two examples are given to illustrate the main results and to show an important difference between the fractional-delay systems with delay-dependent coefficients and the ones with delay-free coefficients from the viewpoint of stability switches.

“Delay Scheduling”: A New Concept for Stabilization in Multiple Delay Systems

2005 ◽  
Vol 11 (9) ◽  
pp. 1159-1172 ◽  
Author(s):  
Nejat Olgac ◽  
Ali Fuat Ergenc ◽  
Rifat Sipahi
Keyword(s):  
Time Delays ◽  
Linear Time ◽  
Complex Problem ◽  
Delay Systems ◽  
Analytical Determination ◽  
Multiple Time ◽  
Control Performance ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
Study Results

A trajectory-tracking problem is considered for a linear time invariant (LTI) dynamics with a fixed control law. However, the feedback line is affected by multiple time delays. The stability of the dynamics becomes a complex problem. It is well known that time-delayed LTI systems may exhibit multiple stable operating zones (which we call pockets) in the space of the delays. Our aim in this paper is to locate and experimentally validate these pockets. For the analytical determination of the pockets we utilize a new methodology, the cluster treatment of characteristic roots (CTCR). The study results in several interesting conclusions. (i) The systems may exhibit better control performance (for instance, faster disturbance rejection) for larger time delays. (ii) Consequently, we propose a unique and interesting utilization of the time delays as agents to enhance the control performance, the delay scheduling technique.

scholarly journals Predicting the secondary dynamic mode interference phenomenon in thermoacoustic instability control

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160182 ◽  
Author(s):  
Umut Zalluhoglu ◽  
Nejat Olgac
Keyword(s):  
Feedback Control ◽  
Helmholtz Resonator ◽  
Dynamic Mode ◽  
Multiple Time ◽  
Interference Phenomenon ◽  
Stabilizing Controller ◽  
Rijke Tube ◽  
Thermoacoustic Instability ◽  
Characteristic Roots ◽  
The Stability

This paper brings a novel mathematical perspective in assessing the rise of the secondary dynamic modes to prominence during the suppression of thermoacoustic instability. This phenomenon is observed by many earlier investigators; however, without a complete analytical reasoning. We consider a Rijke tube with both a passive Helmholtz resonator and an active feedback control to suppress instabilities. The core dynamics is represented as a linear time-invariant multiple time-delay system of neutral type. Parametric stability of the resulting infinite-dimensional dynamics is investigated using a recent analytical tool: cluster treatment of characteristic roots paradigm. This tool reveals the stability outlook of such systems exhaustively and non-conservatively in the parameter space of the system. First, we examine the stability with and without the Helmholtz resonator. We then select an unstable operation for the resonator-mounted Rijke tube, impose a time-delayed integral feedback control over it and reveal the stabilizing controller parameters using the cluster treatment of characteristic roots methodology. When high control gains are inappropriately selected, the new analytical procedure declares how the secondary dynamic modes of the system exhibit instability although the initially unstable mode is now stabilized. All of these stability assessments are cross-validated using experimental results from a laboratory-scale Rijke tube set-up.

Kernel and Offspring Concepts for the Stability Robustness of Multiple Time Delayed Systems (MTDS)

2005 ◽  
Author(s):  
Rifat Sipahi ◽  
Nejat Olgac
Keyword(s):  
Time Delays ◽  
Linear Time ◽  
Characteristic Root ◽  
Test Point ◽  
Invariance Property ◽  
Multiple Time ◽  
Delayed Systems ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
The Stability

A novel treatment for the stability of linear time invariant (LTI) systems with rationally independent multiple time delays is presented. The independence of delays makes the problem much more challenging compared to the systems with commensurate time delays (where the delays have rational relations). It is shown that the imaginary characteristic roots can all be found along a set of curves in the domain of the delays. They are called the “kernel curves”, and it is proven that their number is small and bounded. All possible time delay combinations, which yield an imaginary characteristic root, lie on a curve so called the offspring of the kernel curves within the domain of the delays. We also claim that the root tendencies show a very interesting invariance property as a test point crosses these curves. An efficient, exact and exhaustive methodology results from these outstanding characteristics. It is unique to the new methodology that, the systems under consideration do not have to possess stable behavior for zero delays. Several example case studies are presented, which are prohibitively difficult, if not impossible to solve using any other peer methodology.

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