Self-excited Oscillations and Fuel Control of a Combustion Process in a Rijke Tube

Abstract

This paper treats the control of nonstationary oscillations of the pressure which arise in the combustion process of a Rijke tube. The low order model of a nonstationary combustion process in a Rijke tube with mul tiplicative feedback control function is considered, which is investigated by means of two-DOF coupled van der Pol oscillator. The self-excited oscillations in the combustion process are analysed by means of the Extended Lindstedt-Poincare (EL-P) method with multiple time scales. By applying the EL-P method, it is ascertained, that nonstationary oscillations are almost periodic with amplitude and phase, which are functions of the slow time scale. Without control of the fuel influx, the self excited oscillations can approach limit cycles. By applying the fuel control, the influence of control parameters on the quenching of the self excited oscillations is investigated in a phenomenon of competitive quenching and mutual synchronization with close frequencies and multiple harmonics, respectively.

Modeling and Control of Instabilities in Combustion Processes

“…Ref. [3] it is shown, that phase angles change very little and therefore nonlinear frequencies ω i1 are approximately zero, . It is also shown, that an analytical solution of Eq.…”

“…For solving autonomous system of oscillators, Culick [1] has proposed the method of Krilov-Bogoliubov (KB method). Instead of KB method, an alternative Linstedt-Poincare method with multiple time scales (EL-P method) is proposed in this paper, which has proven very efficient in solving problems of the control of unstable selfexcited oscillations of acoustic pressure in combustion chamber [3]. To demonstrate the usefullness of EL-P method with multiple time scales in classic model of instabilities in a combustion process, that is, to apply EL -P method on Eqs.…”

“…These natural frequencies correspond to two mode shapes in accordance with the experimental findings. Governing equations of the model with control function are generalized van der Pol equations of two modal amplitudes forming the amplitude of pressure oscillations [3]:…”

“…1. Inclusion of low pass filter in block scheme is optional, however, the use of low pass filter dramatically improves properties of the control [3].…”

Second-order sliding-mode differentiators: an experimental comparative analysis using Van der Pol oscillator

“…where x 1 , x 2 ∈ R are state variables, y ∈ R is the output and is the model parameter. Model (3) has been successfully applied in various fields like biomedical engineering [Ryzhii & Ryzhii (2014); Kaplan et al (2008)], control [Puvsenjak et al (2014); Landau et al (2008)], electrical networks [Sinha et al (2016)], etc. Because of its practical importance, state estimation of model ( 3) is a very interesting problem.…”