مدل سازی در مهندسی

مدل سازی در مهندسی

تحلیل وجود چرخه حدی پایدار در یک نوسانگر حرارتی-صوتی دیافراگمی با استفاده از نظریه پایداری لیاپانوف سیستم های غیرخطی دارای آشفتگی

نوع مقاله : مقاله پژوهشی

نویسندگان
1 گروه ریاضی، دانشگاه پیام نور، ص.پ. ،4697-19395 تهران ، ایران
2 گروه مهندسی برق، دانشکده مهندسی برق و کامپیوتر، دانشگاه صنعتی قوچان، قوچان، ایران
چکیده
در این مقاله به تحلیل وجود چرخه حدی پایدار در یک نوسانگر حرارتی-صوتی دیافراگمی با استفاده از نظریه پایداری لیاپانوف سیستم های دینامیکی دارای آشفتگی پرداخته شده است. در این راستا و با کمک نظریه پایداری لیاپانوف، ابتدا معالات دیفرانسیلی دینامیکی حاکم بر نوسانگر حرارتی به صورت معادلات حالت بیان می شوند. سپس برای آنکه سیستم دارای چرخه حدی باشد معادلات حالت خطا به کمک معادلات سیستم و حالتهای مطلوب چرخه حدی نوشته می شوند. در ادامه تابع لیاپانوف جدید برای دینامیک توصیف کننده خطا معرفی و شرایط پایداری مجانبی دینامیک خطا با توجه به این تابع لیاپانوف بررسی می گردد. از شروط ارائه شده شرط اول مربوط به مثبت معین بودن تابع لیاپانوف و شرط دوم مربوط به منفی بودن مشتق این تابع می-باشد. سومین شرط نیز تضمین کننده آنست که بردار حالت خطا در یک کران معین قرار گیرد تا اطمینان یابیم چرخه حدی پایدار مجانبی تحت این شرایط رخ می دهد. از این رو کران بالا و پایین خطا به دست می آیند . همچنین تاثیر برخی پارامترهای مهم فیزیکی سیستم بر روی باند خطای به دست آمده تحلیل شده است. نتایج بدست آمده از این مقاله نشان داد که روش ارائه شده به خوبی توانسته است مهمترین چالش این نوع موتورها یعنی تضمین نوسانات پایدار قبل از ساخت را برطرف سازد.
کلیدواژه‌ها
موضوعات

عنوان مقاله English

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

نویسندگان English

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
چکیده English

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.

کلیدواژه‌ها English

Diaphragm thermo-acoustic oscillator
Nonlinear dynamics
Lyapunov stability
Limit cycle
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دوره 24، شماره 85
در حال تکمیل شدن
تیر 1405
صفحه 169-186

  • تاریخ دریافت 25 فروردین 1404
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