تحلیل پایداری دینامیکی تیرهای FGM بر اساس مدل غیرخطی تیموشنکو

ملک زاده فرد, کرامت; پورموید, علیرضا

doi:10.47176/jcme.42.1.8602

نوع مقاله : مقاله پژوهشی

نویسندگان

¹ دانشگاه صنعتی مالک اشتر

² دانشگاه پدافند هوایی خاتم‌الانبیاء (ص)

https://doi.org/10.47176/jcme.42.1.8602

چکیده

وقوع انواع ناپایداری دینامیکی در سیستم‌های مکانیکی از مهم‌ترین عوامل مختل‌کننده فعالیت در این سازه‌ها است. لذا، مطالعه دقیق ناپایداری دینامیکی در تیرها، به عنوان یکی از اساسی‌ترین ساختارهای مهندسی، از اهمیت بالایی برخوردار است. در این مقاله، مساله ناپایداری دینامیکی تیرهای ساخته شده از مواد مدرج یا هوشمند تابعی (FGM) مطالعه شده است. برای این منظور، تئوری تیر برشی مرتبه اول یا تیموشنکو با اثرات غیرخطی بودن هندسی لحاظ شده است. به این ترتیب، مدل پیشنهادی قابلیت تعیین رفتار مکانیکی تیرهای نازک و ضخیم را داراست. با در نظر گرفتن انواع توابع انرژی سیستم و پیاده‌سازی اصل همیلتون، معادلات حاکم بر مساله به همراه انواع شرایط مرزی متداول به دست آمده است. روش تربیع دیفرانسیلی (DQM) به عنوان یکی از شناخته‌شده‌ترین روش‌های حل عددی مساله به کار گرفته شده و معادلات غیرخطی دیفرانسیلی با مشتقات جزیی به صورت معادل به شکل معادلات دیفرانسیلی با مشتقات معمولی نوشته می‌شوند. سپس با در نظر گرفتن پاسخ‌های هارمونیک برای سیستم، معادلات دیفرانسیلی به مجموعه‌ای از معادلات غیرخطی جبری تبدیل شده‌اند. در انتها، به منظور مطالعه پارامترهای اساسی، مثال‌های عددی مختلفی ارائه شده است. نتایج عددی حاصل با مراجع مقایسه شده و به این ترتیب اعتبار فرمول‌بندی ارائه شده و روش حل موجود مشخص شده است. همچنین مطالعه مقایسه‌ای میان مدل‌های سینماتیک خطی و غیرخطی نشان می‌دهد که اهمیت غیرخطی بودن هندسی مدل کاملاً چشمگیر است.

کلیدواژه‌ها

20.1001.1.22287698.1402.42.1.4.3

عنوان مقاله [English]

Dynamic Stability Analysis of FGM Beams Based on the Nonlinear Timoshenko Model

نویسندگان [English]

Keramat Malakzadeh Fard ¹
Alireza Pourmoayed ²

چکیده [English]

Various types of dynamic instabilities in mechanical systems are one of the most important disruptive factors in such structures. Therefore, an accurate study of dynamic instability in beams, as one of the fundamental engineering structures, is of great importance. In this paper, dynamic instability problem of beams made of Functionally Graded Materials (FGM) is investigated. For this purpose, the first-order shear deformation (or the Timoshenko) beam theory with the effects of geometric nonlinearity is considered. Thus, the proposed model has the ability to determine mechanical behavior of thin and thick beams. By considering the energy functions of the system, and implementing the Hamilton’s principle, the governing equations are obtained along with different types of common boundary conditions. The Differential Quadrature Method (DQM), as one of the best-known numerical methods, is used. The nonlinear partial differential equations are written in the form of equivalent ordinary differential equations. Then, considering the harmonic responses for the system, the differential equations are converted to a set of nonlinear algebraic equations. Finally, in order to study the important parameters, various numerical examples are provided. The obtained numerical results are compared with the literature and thus, the validity of the presented formulation and solution methodology is revealed. Also, a comparative study between linear and nonlinear kinematic models shows that the importance of geometric nonlinearity of the model is quite significant.

کلیدواژه‌ها [English]

Dynamic instability
functionally graded material
nonlinear kinematics
Timoshenko beam theory
differential quadrature solution method

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