A Stable Limit Cycle Existence Analysis in a Diaphragm Thermos-Acoustic Oscillator Using Lyapunov Stability Theorem of Perturbed Systems

Teymuri, Azam; Heydari, Aghileh; Ramezani-al, Mohammad Reza

doi:10.22075/jme.2025.37356.2837

A Stable Limit Cycle Existence Analysis in a Diaphragm Thermos-Acoustic Oscillator Using Lyapunov Stability Theorem of Perturbed Systems

Document Type : Research Paper

Authors

Azam Teymuri ¹

Aghileh Heydari ¹

Mohammad Reza Ramezani-al ²

¹ Department of Mathematics, Payam Noor University, Tehran, Iran

² Department of Electrical Engineering, Facualty of Electrical and Computer Engineering, Quchan University of Technology, Quchan, Iran

10.22075/jme.2025.37356.2837

Abstract

In this paper, the existence of a stable limit cycle in a diaphragm thermos-acoustic oscillator is analyzed using the Lyapunov stability theorem of perturbed dynamic systems. In this regard, first the dynamic differential equations of the thermos-acoustic oscillator are written as the state equations. Then, in order to system have a limit cycle, the error equations are extracted using the system equations and the desired states of the limit cycle. Next, a new Lyapunov function is introduced to asymptotic stability analysis of the error dynamics. Three conditions are examined, first condition of stability related to the positive definiteness of the Lyapunov function and the second condition related to the negativity of the derivative of this function. Third condition also guarantees that the error state is located within a certain bound to ensure that an asymptotic stable limit cycle occurs under these conditions. Besides, the upper and lower bounds of the error are obtained. Also, the effect of some important physical parameters of the system on the obtained error's bounds has been analyzed. The results obtained showed that the presented method has been able to successfully solve the most important challenge of this type of thermal oscillator, namely ensuring selecting the proper parameters for a thermo-acoustic oscillations.

Keywords

Diaphragm thermo-acoustic oscillator

Nonlinear dynamics

Lyapunov stability

Limit cycle

Subjects

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