ارتعاشات اجباری غیرخطی صفحۀ مدرج تابعی مگنتو-الکترو-الاستیک مستطیلی‌شکل براساس تئوری تغییرشکل برشی مرتبۀ سوم

نوع مقاله : مقاله پژوهشی

نویسندگان

• رامین منتشلو ¹
• علیرضا شوشتری ²

¹ دانشگاه بوعلی سینا

² دانشگاه بوعلی سینا همدان

10.22067/fum-mech.v30i2.66477

چکیده

در این مقاله ارتعاشات اجباری عرضی غیرخطی تحت نیروی تحریک هارمونیک جانبی یک صفحۀ مدرج تابعی مگنتو-الکترو-الاستیک مستطیلی‌شکل ساخته شده از پیزوالکتریک BaTiO3 و مگنتواستریکتیو CoFe2O4، با شرایط مرزی تکیه‌گاه ساده بررسی شده است. تغییرات خواص از پایین به بالا فرض شده است، یعنی سطح بالای ورق غنی از پیزوالکتریک است، در حالی که سطح پایین ورق غنی از مگنتواستریکتیو است و خواص مواد در جهت ضخامت مطابق قانون توزیع توانی درجه‌بندی شده است. معادلات حرکت براساس تئوری برشی مرتبه سوم رِدی به‌دست آمده­اند. همچنین معادلات ماکسول در حالت الکترواستاتیک و مگنتواستاتیک برای مدل‌سازی رفتار الکتریکی و مغناطیسی به‌کار برده شده­اند. سپس معادلات دیفرانسیل جزئی غیرخطی به‌دست آمده با استفاده از روش گلرکین به پنج معادلۀ دیفرانسیل معمولی غیرخطی کوپل شده تبدیل شده­اند. این معادلات دیفرانسیل معمولی غیرخطی به یک معادلۀ دیفرانسیل که شامل ترم­های غیرخطی سفتی و اینرسی و ترم­های غیرخطی مرتبه دو و سه است کاهش یافته­اند. برای حل معادلۀ به‌دست آمده، از تئوری اغتشاشات استفاده شده است و سپس نتایج به‌دست آمده برای فرکانس طبیعی خطی با نتایج موجود بـرای ورق­های همسانگرد و ورق­هـای پیزوالاستیک همسانگرد مقایسه شده­اند و مطابقت خوبی میان نتایج این پژوهش و نتایجی که قبلاً منتشر شده­اند مشاهده گردیده­است. در تحلیل ارتعاشات اجباری، تشدیدهای اولیه، سوپرهارمونیک مطالعه شده­است و معادلۀ پاسـخ فرکانسـی آن به‌دست آمده است. همچنین باتوجه به اهمیت تشدید اولیه، پایداری آن نیز بررسی شده است. نیروی تحریک خارجی در این حالت، نیروی تحریک مکانیکی با دامنـۀ ثابـت و به‌صورت هارمونیک در حوزۀ زمان در نظر گرفته شده است.

کلیدواژه‌ها

• ورق هوشمند مدرج تابعی
• تئوری تغییر شکل برشی مرتبه سوم
• قوانین گاوس
• تئوری اغتشاشات
• ارتعاش غیرخطی اجباری

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عنوان مقاله [English]

Nonlinear Forced Vibration of Functionally Graded Magneto-Electro-Elastic Rectangular Plate based on the Third Order Shear Deformation Theory

نویسندگان [English]

• Ramin Mantashloo ¹
• alireza shooshtari ²

¹ Bu-Ali Sina University

² Bu-Ali Sina University

چکیده [English]

In this paper, nonlinear vibration of the functionally graded rectangular plates made of piezoelectric BaTiO3 and magnetostrictive CoFe2O4, whit simply supported boundary condition has been investigated. It is assumed that the composition is varied from the bottom surfaces to top surface, i.e., the top surface of the plate is piezoelectric-rich, whereas the bottom surface is magnetostrictive-rich. In addition, material properties are graded along the thickness according to volume fraction power-law distribution. Based on the Reddy’s third-order share deformation plate theory, the governing equations of motion, whereas Maxwell equations for electrostatics and magnetostatics are used to model the electric and magnetic behavior. Then, the nonlinear partial differential equations of motion are transformed into five coupled nonlinear ordinary differential equations by using the Galerkin method. Afterward, the obtained coupled ordinary differential equations are reduced to a single nonlinear differential equation which include nonlinear inertia and stiffness terms with quadratic and cubic nonlinear terms. A perturbation method is used to solve the equation of motion analytically. The results for natural frequency are compared with the available results for isotropic, laminated and piezo-laminated plates and good agreement is found between the results of present study with the results of previously published papers. In the forced vibration, primary, super-harmonic resonances are studied and the frequency response equation has been obtained. Because of the importance of the primary resonance, the stability of the steady-state motion is investigated for the primary resonance, The applied external force is assumed to be harmonic in time with a constant amplitude.

کلیدواژه‌ها [English]

• Functionally graded smart plate
• Third Order Shear Deformation Theory
• Gauss’s laws
• perturbation method
• force vibration

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