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توسعه روش اغتشاشات- هوموتوپی به منظور بررسی ارتعاشات غیرخطی تیر مدرج تابعی متخلخل تحت بارگذاری خارجی

نوع مقاله : مقاله پژوهشی

نویسندگان

1 استادیار ، گروه مهندسی مکانیک ، دانشگاه شهید مدنی آذربایجان ، تبریز

2 دکتری ، گروه مهندسی مکانیک ، دانشگاه تبریز ، تبریز

10.22068/jstc.2020.116889.1604

چکیده

در مقاله حاضر، ارتعاشات غیرخطی عرضی با دامنه بزرگ تیر مدرج تابعی متخلخل واقع بر بستر وینکلر تحت بارگذاری خارجی به صورت تحلیلی مورد مطالعه قرار داده می‌شود. خواص تیر مدرج تابعی متخلخل مطابق با قانون توانی به طور پیوسته در سطح مقطع تیر تغییر می‌کند. معادله دیفرانسیل حاکم بر حرکت تیر اولر- برنولی با بکارگیری اصل هامیلتون و با فرض غیرخطینگی ون کارمن به دست می‌آید. با اعمال روش گالرکین و با درنظر گرفتن شرایط مرزی دو سر گیردار غیرمتحرک، همچنین با فرض بار خارجی یکنواخت خارجی، معادله دیفرانسیل با مشتقات جزئی، تبدیل به معادله دیفرانسیل معمولی می‌شود. به دلیل بزرگ بودن ضریب جمله غیرخطی و نیز به دلیل ناهمگن بودن معادله غیرخطی حاصل، روش‌های کلاسیک اغتشاشات قادر به حل مساله نمی-باشند. به منظور حل، روش اغتشاشات- هوموتوپی اصلاح‌شده، توسعه داده می‌شود. در ادامه برای صحه‌گذاری، نتایج پاسخ زمانی به دست آمده از روش اغتشاشات- هوموتوپی توسعه‌یافته با نتایج عددی به ازای دامنه‌ها و فرکانس-های تحریک مختلف، مقایسه می-شود. همچنین نتایج به دست آمده برای مقدار افزایش فرکانس غیرخطی تیر مدرج تابعی متخلخل دوسرگیردار با روش حاضر و روش-های موجود در ادبیات فن مورد مقایسه قرار
می‌گیرد. مقایسه نتایج نشان می-دهد که روش ارائه شده از دقت بسیار خوبی برخوردار است. در نهایت، منحنی‌های پاسخ فرکانسی تیر به ازای مقادیر مختلف شاخص توانی کسر حجمی و نیز کسر حجمی تخلخل، ارائه شده و مورد بررسی قرار می‌گیرد.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Developing homotopy perturbation method to investigate the nonlinear vibration of a Porous FG-Beam subjected to the external excitation

نویسندگان [English]

  • Masoud Minaei 1
  • Vahid Arab Maleki 2

1 Department of Mechanical Engineering, Azarbaijan Shahid Madani University, Tabriz, Iran

2 Department of Mechanical Engineering, University of Tabriz, Tabriz, Iran

چکیده [English]

In this paper large amplitude vibration analysis of a porous FG-beam rested on a Winkler foundation and subjected to a harmonic loading is analytically investigated. The material properties of the porous FG beam are assumed to vary continuously according to a simple power law. Employing Von Karman’s geometric nonlinearity, implementing the Galerkin’s method and assuming doubly clamped immovable end boundary conditions, the governing nonlinear partial differential equation is reduced to a nonlinear ODE. Because of the large coefficient of the nonlinear term and due to existence of the external harmonic loading effect, none of the traditional perturbation methods leads to a valid solution. So, in order to solve this nonlinear nonhomogenous equation, the modified homotopy perturbation method is developed and it is called developed homotopy perturbation method (DHPM). For validating, the time response results obtained by DHPM and numerical methods are compared for various values of excitation amplitudes and frequencies. The increasing of nonlinear frequency obtained by DHPM with those of existence literature revealed a good agreement with a desired accuracy. Finally, the frequency response curves are presented for different values of volume fraction exponent together with porous volume fraction and the effects of nonlinearities on the frequency response curves are discussed in detail.

کلیدواژه‌ها [English]

  • Nonlinear Forced Vibration
  • Porous FG-Beam
  • Developing Homotopy Perturbation Method
  • Winkler Foundation
مراجع
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