Nonlinear Phenomena in Thermoacoustic Systems With Premixed Flames

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Karthik Kashinath ◽  
Santosh Hemchandra ◽  
Matthew P. Juniper
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycle ◽  
Release Rate ◽  
Time Domain ◽  
Limit Cycles ◽  
Integral Relation ◽  
Single Mode ◽  
Rate Of Change ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

Nonlinear analysis of thermoacoustic instability is essential for the prediction of the frequencies, amplitudes, and stability of limit cycles. Limit cycles in thermoacoustic systems are reached when the energy input from driving processes and energy losses from damping processes balance each other over a cycle of the oscillation. In this paper, an integral relation for the rate of change of energy of a thermoacoustic system is derived. This relation is analogous to the well-known Rayleigh criterion in thermoacoustics, however, it can be used to calculate the amplitudes of limit cycles and their stability. The relation is applied to a thermoacoustic system of a ducted slot-stabilized 2-D premixed flame. The flame is modeled using a nonlinear kinematic model based on the G-equation, while the acoustics of planar waves in the tube are governed by linearized momentum and energy equations. Using open-loop forced simulations, the flame describing function (FDF) is calculated. The gain and phase information from the FDF is used with the integral relation to construct a cyclic integral rate of change of energy (CIRCE) diagram that indicates the amplitude and stability of limit cycles. This diagram is also used to identify the types of bifurcation the system exhibits and to find the minimum amplitude of excitation needed to reach a stable limit cycle from another linearly stable state for single-mode thermoacoustic systems. Furthermore, this diagram shows precisely how the choice of velocity model and the amplitude-dependence of the gain and the phase of the FDF influence the nonlinear dynamics of the system. Time domain simulations of the coupled thermoacoustic system are performed with a Galerkin discretization for acoustic pressure and velocity. Limit cycle calculations using a single mode, along with twenty modes, are compared against predictions from the CIRCE diagram. For the single mode system, the time domain calculations agree well with the frequency domain predictions. The heat release rate is highly nonlinear but, because there is only a single acoustic mode, this does not affect the limit cycle amplitude. For the twenty-mode system, however, the higher harmonics of the heat release rate and acoustic velocity interact, resulting in a larger limit cycle amplitude. Multimode simulations show that, in some situations, the contribution from higher harmonics to the nonlinear dynamics can be significant and must be considered for an accurate and comprehensive analysis of thermoacoustic systems.

Nonlinear Phenomena in Thermoacoustic Systems With Premixed Flames

2012 ◽  
Author(s):  
Karthik Kashinath ◽  
Santosh Hemchandra ◽  
Matthew P. Juniper
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycle ◽  
Release Rate ◽  
Time Domain ◽  
Limit Cycles ◽  
Integral Relation ◽  
Single Mode ◽  
Rate Of Change ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

Nonlinear analysis of thermoacoustic instability is essential for prediction of frequencies, amplitudes and stability of limit cycles. Limit cycles in thermoacoustic systems are reached when the energy input from driving processes and energy losses from damping processes balance each other over a cycle of the oscillation. In this paper an integral relation for the rate of change of energy of a thermoacoustic system is derived. This relation is analogous to the well-known Rayleigh criterion in thermoacoustics, but can be used to calculate the amplitudes of limit cycles, as well as their stability. The relation is applied to a thermoacoustic system of a ducted slot-stabilized 2-D premixed flame. The flame is modelled using a nonlinear kinematic model based on the G-equation, while the acoustics of planar waves in the tube are governed by linearised momentum and energy equations. Using open-loop forced simulations, the flame describing function (FDF) is calculated. The gain and phase information from the FDF is used with the integral relation to construct a cyclic integral rate of change of energy (CIRCE) diagram that indicates the amplitude and stability of limit cycles. This diagram is also used to identify the types of bifurcation the system exhibits and to find the minimum amplitude of excitation needed to reach a stable limit cycle from another linearly stable state, for single-mode thermoacoustic systems. Furthermore, this diagram shows precisely how the choice of velocity model and the amplitude-dependence of the gain and the phase of the FDF influence the nonlinear dynamics of the system. Time domain simulations of the coupled thermoacoustic system are performed with a Galerkin discretization for acoustic pressure and velocity. Limit cycle calculations using a single mode, as well as twenty modes, are compared against predictions from the CIRCE diagram. For the single mode system, the time domain calculations agree well with the frequency domain predictions. The heat release rate is highly nonlinear but, because there is only a single acoustic mode, this does not affect the limit cycle amplitude. For the twenty-mode system, however, the higher harmonics of the heat release rate and acoustic velocity interact resulting in a larger limit cycle amplitude. Multi-mode simulations show that in some situations the contribution from higher harmonics to the nonlinear dynamics can be significant and must be considered for an accurate and comprehensive analysis of thermoacoustic systems.

Sensitivity Enhancement of Cantilever-Based Sensors Using Feedback Delays

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Calvin Bradley ◽  
Mohammed F. Daqaq ◽  
Amin Bibo ◽  
Nader Jalili
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Stable Limit Cycle ◽  
Sensitivity Enhancement ◽  
Design Parameters ◽  
Feedback Signal ◽  
Stable Limit ◽  
Theoretical Understanding ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

This paper entails a novel sensitivity-enhancement mechanism for cantilever-based sensors. The enhancement scheme is based on exciting the sensor at the clamped end using a delayed-feedback signal obtained by measuring the tip deflection of the sensor. The gain and delay of the feedback signal are chosen such that the base excitations set the beam into stable limit-cycle oscillations as a result of a supercritical Hopf bifurcation of the trivial fixed points. The amplitude of these limit-cycles is shown to be ultrasensitive to parameter variations and, hence, can be utilized for the detection of minute changes in the resonant frequency of the sensor. The first part of the manuscript delves into the theoretical understanding of the proposed mechanism and the operation concept. Using the method of multiple scales, an approximate analytical solution for the steady-state limit-cycle amplitude near the stability boundaries is obtained. This solution is then utilized to provide a comprehensive understanding of the effect of small frequency variations on the limit-cycle amplitude and the sensitivity of these limit-cycles to different design parameters. Once a deep theoretical understanding is established, the manuscript provides an experimental study to investigate the proposed concept. Experimental results demonstrate orders of magnitude sensitivity enhancement over the traditional frequency-shift method.

Reduction of Limit Cycle Amplitude in the Presence of Backlash

1999 ◽  
Vol 121 (2) ◽  
pp. 278-284 ◽  
Author(s):  
Ronen Boneh ◽  
Oded Yaniv
Keyword(s):  
Nonlinear Control ◽  
Limit Cycle ◽  
Closed Loop ◽  
Controller Design ◽  
Linear Time ◽  
Design Technique ◽  
Feedback Systems ◽  
Mass System ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

The majority of feedback systems driven by an electric motor can be represented by a two-mass system connected via a flexible drive element. Owing to the presence of backlash, the closed-loop performance such as precision speed, position and force control that can be achieved using a linear time invariant controller is limited, and it is expected that a nonlinear control would be superior. In this paper a nonlinear control structure is proposed and a systematic design technique presented. The advantages of the proposed design technique are: (i) It is robust to plant and backlash uncertainty; (ii) it is quantitative to specifications on the maximum limit cycle amplitude; and (iii) the closed loop is superior to a linear controller design both in lower bandwidth and in lower limit cycle amplitude. A design example is included.

Nonlinear Multi-Modal Panel Flutter Oscillations at Low Supersonic Speeds

2014 ◽  
Author(s):  
Vasily Vedeneev ◽  
Anastasia Shishaeva ◽  
Konstantin Kuznetsov ◽  
Andrey Aksenov
Keyword(s):  
Numerical Simulation ◽  
Limit Cycle ◽  
Time Domain ◽  
Internal Resonance ◽  
Gas Flow ◽  
Single Mode ◽  
Panel Flutter ◽  
Limit Cycle Oscillations ◽  
Plate Deformation ◽  
Coupled Mode

In this paper aeroelastic instability of a plate in a gas flow is investigated by direct time-domain numerical simulation. Plate deformation and gas flow are simulated in solid and fluid codes, respectively, with direct coupling between these codes. A series of simulations under different parameters has been conducted. Three types of the plate response have been observed: stability, static divergence and flutter. Depending on Mach number, two types of flutter were detected: single mode flutter and coupled mode flutter. At M = 1.8, a good correlation between the present study and the piston theory for coupled mode flutter has been obtained. At lower M, from 1 to 1.6, single mode flutter in 1st, 2nd and higher modes has been observed. Amplitudes and frequencies of flutter limit cycle oscillations have been studied. It is shown that limit cycle oscillations can occur in form of pure one-mode oscillations, or include 1:2 internal resonance, when fluttering mode excites another mode. In the region of Mach numbers from 1.3 to 1.5, where several plate modes are simultaneously unstable, transition from periodic to quasi-chaotic flutter oscillations occurs.

Limit Cycle Behavior of a Flexible Truck

1987 ◽  
Vol 54 (4) ◽  
pp. 930-934 ◽  
Author(s):  
A. M. Whitman ◽  
J. E. Molyneux
Keyword(s):  
Limit Cycle ◽  
Critical Speed ◽  
Perturbation Procedure ◽  
Cycle Amplitude ◽  
Creep Force ◽  
Limit Cycle Amplitude ◽  
Limit Cycle Behavior

We calculate the variation in critical speed of a flexible truck as a function of limit cycle amplitude and truck parameters (i.e., shear and bending stiffnesses, and truck geometry), by means of a perturbation procedure. We find that the creep force nonlinearity is dominant, and that it can cause the nonlinear critical speed to be either lower or higher than the linear critical speed, depending on the values of the two stiffnesses.

Algorithm for Controlling Spillback from Ramp Meters

1996 ◽  
Vol 1554 (1) ◽  
pp. 162-171 ◽  
Author(s):  
Robert L. Gordon
Keyword(s):  
Limit Cycle ◽  
Control Algorithm ◽  
Arrival Rate ◽  
Sampling Period ◽  
High Amplitude ◽  
Ramp Metering ◽  
Control Algorithms ◽  
Considerable Distance ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

The queue control algorithms that are commonly used by ramp-metering controllers often permit the queue to extend a considerable distance upstream of the queue detector on the ramp. This detector senses the presence of a queue with the intent of preventing queue spillback onto the surface street or intersection that is located upstream of the ramp. Queue control is usually obtained by switching between the desired metering rate and a higher metering rate. In some cases metering is terminated. This type of limit cycle control often results in considerable variation in the length of the queue, even for a constant average vehicle arrival rate at the ramp. This high-amplitude limit cycle operation is indicated, by simulation, to result from the relatively long detector sampling and computation period (typically 1 min) that is commonly used. A control algorithm using a 10-sec sampling period is indicated to substantially reduce the limit cycle amplitude. Performance is further improved when the control algorithm anticipates the buildup of the queue over the spillback detector.

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