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.

Analytical and Experimental Study on Passive Stabilization of Thermoacoustic Dynamics in a Rijke Tube

2017 ◽  
Vol 140 (2) ◽  
Author(s):  
Umut Zalluhoglu ◽  
Nejat Olgac
Keyword(s):  
Linear Time ◽  
Helmholtz Resonator ◽  
Delay Systems ◽  
Space Representation ◽  
Multiple Time ◽  
Mathematical Tool ◽  
Rijke Tube ◽  
Thermoacoustic Instability ◽  
Characteristic Roots ◽  
Passive Stabilization

This paper deals with passive stabilization of thermoacoustic dynamics in a Rijke tube using a Helmholtz resonator. Thermoacoustic instabilities result from the dynamic coupling between the heat release and pressure in a chamber. Helmholtz resonators are used akin to vibration absorbers to suppress unwanted pressure oscillations in such structures and prevent instabilities. The first contribution of the paper is a state-space representation of the thermoacoustic dynamics for the resonator-mounted Rijke tube. This relationship happens to be in the class of linear time invariant, neutral multiple time delay systems (LTI-NMTDS). Then, benefiting from the cluster treatment of characteristic roots (CTCR) paradigm, we investigate the effect of resonator location on suppression of thermoacoustic instability. CTCR is a mathematical tool that determines the stability of LTI-NMTDS exhaustively and nonconservatively in the parameter space of the system. This capability provides a novel tool for the futuristic design concepts of combustors. These analytically obtained findings are also supported with experimental results from a laboratory-scale Rijke tube. In addition, a conceptual case study is presented where the stabilizing contributions of the resonator to the dynamics are investigated under strong thermoacoustic coupling.

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.

One-dimensional model for the Rijke tube

1989 ◽  
Vol 202 ◽  
pp. 83-96 ◽  
Author(s):  
C. Nicoli ◽  
P. Pelcé
Keyword(s):  
Heat Transfer ◽  
Transfer Function ◽  
Dimensional Model ◽  
Unsteady Heat Transfer ◽  
Order Unity ◽  
Rijke Tube ◽  
Thermoacoustic Instability ◽  
One Dimensional ◽  
The Stability ◽  
Stability Limits

We develop a simple model in which longitudinal, compressible, unsteady heat transfer between heater and gas is computed in the small-Mach-number limit. This calculation is used to determine the transfer function of the heater, which plays an important role in the stability limits of the thermoacoustic instability of the Rijke tube. The transfer function is determined analytically in the limit of small expansion parameter γ, and numerically for γ of order unity. In the case ρμ/cp = constant, an analytical solution can be found.

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.

Refined Stability Conditions for Linear Uncertain Systems With Multiple Time-Varying Delays

2006 ◽  
Vol 129 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Chih-Peng Huang
Keyword(s):  
Uncertain Systems ◽  
State Feedback ◽  
Multiple Time ◽  
Time Varying ◽  
Closed Loop System ◽  
Stabilizing Controller ◽  
Time Varying Delay ◽  
The Stability ◽  
Varying Delay ◽  
Explicit Criteria

This paper mainly proposes distinct criteria for the stability analysis and stabilization of linear uncertain systems with time-varying delays. Based on the Lyapunov theorem, a sufficient condition of the unforced systems with single time-varying delay is first derived. By involving a memoryless state feedback controller, the condition will be extended to treat with the resulting closed-loop system. These explicit criteria can be reformulated in LMIs forms, so we will readily verify the stability or design a stabilizing controller by the current LMI solver. Furthermore, the considered systems with multiple time-varying delays are similarly addressed. Numerical examples are given to demonstrate that the proposed approach is effective and valid.

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.

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.

Stability of a Consensus Protocol for Second Order Agents With Multiple Time Delays

2011 ◽  
Author(s):  
Rudy Cepeda-Gomez ◽  
Nejat Olgac
Keyword(s):  
Information Exchange ◽  
Time Delays ◽  
Second Order ◽  
Multiple Time ◽  
State Transformation ◽  
Consensus Protocol ◽  
Position Information ◽  
Characteristic Roots ◽  
Velocity Information ◽  
The Stability

In this study we consider the consensus problem for a group of second order agents interacting under a fixed, undirected communication topology. Communication lines are affected by two rationally independent delays. The first delay is assumed to be in the position information channels whereas the second one is in the velocity information exchange. The delays are assumed to be uniform throughout the entire network. We first reduce the complexity of the problem, by performing a state transformation that allows the decomposition of the characteristic equation of the system into a set of second order factors. The stability of the resulting subsystems is analyzed exactly and exhaustively in the domain of the time delays using the Cluster Treatment of Characteristic Roots (CTCR) paradigm. CTCR is a recent method which declares the stability features of the system for any composition of the time delays. Example cases are provided to verify the analytical conclusions.

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