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The Helmholtz method is developed to predict the self-excited thermoacoustic instabilities in a gas turbine combustor, combining flame describing functions, the measured damping rates under the firing condition, and the non-uniform spatial distributions of the physical parameters. The impact of the hydrodynamic and geometrical parameters on the thermoacoustic instabilities is investigated. The measured damping rates show lower values under a hot condition compared with those in a cold state. The experimental results indicate that the relative errors of the predicted eigenfrequencies and the velocity fluctuation levels are below 10%. The pressure amplitude decreases and the phase increases in the axial direction, indicating a typical 1/4-wavelengh mode. At a higher equivalence ratio, the mode shape in the axial direction becomes steeper due to the elevated fluctuation amplitude at the pressure antinode after enhancing the thermal power. When the air flow rate increases, the discrepancies between the pressure shape on the flame tube side and that on the plenum side are reduced. The velocity fluctuation level increases as the combustor length increases at a constant damping rate. In fact, the velocity fluctuation level first increases and then declines, caused by more significant damping rates when employing longer flame tubes. Self-excited thermoacoustic instabilities can be well predicted using the proposed method.

This article describes recent progress on premixed flame dynamics interacting with acoustic waves. Expressions are derived to determine the stability of combustors with respect to thermoacoustic oscillations. The validity of these expressions is general, but they are illustrated in laminar systems. Laminar burners are commonly used to elucidate the response of premixed flames to incoming flow perturbations, highlight the role of acoustic radiation in their stability, identify modes associated with thermoacoustic intrinsic instabilities and decipher the leading mechanisms in annular systems with multiple injectors. Many industrial devices also operate in a laminar premixed mode such as, for example, domestic gas boilers and heaters equipped with matrix burners for material processing in which unconfined flames are stabilized at one extremity of the system. This article proposes a systematic approach to determine the stability of all these systems with respect to thermoacoustic oscillations by highlighting the key role of the burner impedance and the flame transfer function (FTF). This transfer function links in frequency space incoming flow perturbations to heat release rate disturbances. This concept can be used in the turbulent flame case as well. Weakly nonlinear stability analysis can also easily be conducted by replacing the FTF by a flame describing function in the expressions derived in this work. The response of premixed flames to harmonic mixture compositions and flow-rate perturbations is then revisited and the main parameters controlling the FTF are described. A theoretical framework is finally developed to reduce the system thermoacoustic sensitivity by tailoring the FTF.

In this paper, the behavior of a fractional order Van der Pol-like oscillator is investigated using a describing function method. A parametric function for the boundary between oscillatory and nonoscillatory regions of this system is extracted. The analytical results are evaluated by numerical simulations which demonstrate sufficient reliability of the proposed analyzing method.

The design of robust limit-cycle controllers is introduced for autonomous systems with separable SISO nonlinearities. The objective is to design a controller to secure specified robust oscillation amplitude and frequency. The method consists of quasi-linearization of the nonlinear element via a Describing Function (DF) approach and then shaping the loop to reach desired limit-cycle characteristics. As the DF method is used, loop shaping takes place in the Nyquist plot. An example is given to illustrate the robustness of the controlled system to uncertainties in the linear subsystem model.

Combustion instabilities in annular systems raise fundamental issues that are also of practical importance to aircraft engines and ground-based gas turbine combustors. Recent studies indicate that the injector plays a significant role in the stability of combustors by defining the flame dynamical response and setting the inlet impedance of the system. The present investigation examines the effects of combinations of injectors of two different types ($U$ and $S$) on thermoacoustic instabilities in a laboratory-scale annular combustor and compares different circumferential staging strategies. The combustor operates in a stable fashion when all injection units belong to the $S$-family, but exhibits large amplitude pressure oscillations when all these units are of the $U$-type. When the system comprises a mix of $U$- and $S$-injectors, it is possible to determine the number of $S$-injectors leading to stable operation. For a fixed proportion of $U$- and $S$-injectors, some arrangements give rise to stable operation while others do not. Results also show that introducing symmetry-breaking elements affects the system's modal dynamics. These experimental observations are interpreted in an acoustic energy balance framework used to derive an expression for the growth rate as a function of the describing functions of the flames formed by the different injectors and their respective azimuthal locations. Growth rates are determined for the different configurations and used to explain the various observations, estimate the system damping rate and predict the location of the nodal line when the standing mode prevails.

The paper presents a simple, systematic and novel graphical method which uses computer graphics for prediction of limit cycles in two dimensional multivariable nonlinear system having rectangular hysteresis and backlash type nonlinearities. It also explores the avoidance of such self-sustained oscillations by determining the stability boundary of the system. The stability boundary is obtained using simple Routh Hurwitz criterion and the incremental input describing function, developed from harmonic balance concept. This may be useful in interconnected power system which utilizes governor control. If the avoidance of limit cycle or a safer operating zone is not possible, the quenching of such oscillations may be done by using the signal stabilization technique which is also described. The synchronization boundary is laid down in the forcing signal amplitudes plane using digital simulation. Results of digital simulations illustrate accuracy of the method for 2×2 systems.

A first model developed by the authors for the study of pathological oscillations in Parkinson's disease is reviewed. This provides conditions for onset and extinction of almost sinusoidal oscillations. The describing function approach for use in such a situation is introduced and it is shown that the two describing functions used here may be “composed” into a single equivalent describing function. Conditions at onset or quenching of oscillation are noted, but wider applications have yet to be developed.

Owing to limited computational resources and the very large difference between electrical and thermal time constants, simulation for resonantly operated DC–DC converters using physical or ideal switching model is very time-consuming or impossible. In this contribution, a novel thermal–electrical averaging model of resonantly operated DC–DC converters is proposed to reduce the computational burden and time consumption. The thermal–electrical model is based on extended describing function method. All required parameters for the thermal model can be obtained from the datasheets or on the website. The thermal–electrical models are applied in series–parallel resonant converters with LC-type and C-type output filters as examples, and verified by comparison with two standard simulation tools. The thermal–electrical modelling procedure proposed in this contribution can of course also be applied for other types of resonant converters, such as series resonant converter, parallel resonant converter and multi-phase resonant converter.

A new procedure for the design of non-linear proportional and rate feedback controllers based on a previously proposed concept is developed. The procedure is to obtain the sinusoidal-input describing function models of the system, followed by determination of the amplitude-dependent rate feedback gains. Then, this amplitude-dependent gain function is inverted to obtain the non-linear function describing a non-linear rate feedback gain. Then, the amplitude-dependent proportional gains that desensitize the overall open-loop system are determined via optimization. Finally, the obtained amplitude-dependent proportional gains are inverted to obtain the actual non-linear function describing the required non-linear behaviour of the proportional gain. Bounded-input–bounded-output stability is demonstrated by successful generation of the closed-loop describing function models, and the procedure is applied to a servo problem; the results are compared with four other non-linear controller design procedures that were reported in the open literature.

This paper discusses a simple method for analyzing FLC in frequency domain based on describing function. Since nonlinear characteristics of FLC make it difficult FLC analysis, it usually requires a big deal of trial-and-error procedures based on computer simulation. The proposed method is simple and easy to understand, because it is based on the Nyquist stability criterion used to analyze absolute and relative stability, phase and gain margin of a linear system. To linearize in frequency domain, a describing function for FLC is derived by using a piecewise linearization of the FLC response plot. This describing function is represented as a function of magnitude of input sinusoid and nonlinear parameters x 1 and x 2 which change consequence fuzzy variables and nonlinearity of FLC. The describing function is redefined without the magnitude of sinusoid input because maximum values of the describing function can explain the stability of the system. This redefined describing function is used to get minimum stability characteristic, an absolute stability, phase margin and gain margin, of FLC. Using this function, we can explicitly figure out various characteristic of FLC according to x 1 and x 2 in frequency domain. In this work, we suggest a minimum phase margin (MPM) and a minimum gain margin (MGM) for FLC which can be used to determine whether the system is stable or not and how stable it is. For simplicity, we use one-input FLC with three rules. For various nonlinear response of FLC, changing fuzzy variables of a consequence membership function is used. Simulation results show that these parameters are effective in analyzing FLC.