پاسخ تحلیلی ارتعاش آزاد غیرخطی نانو ورق مستطیلی با شرایط مرزی مختلف با استفاده از تئوری غیرمحلی

نوع مقاله : مقاله پژوهشی

نویسندگان

1 دانشجوى دکترا، مهندسى مکانیک، دانشگاه یاسوج

2 استادیار ، مهندسى مکانیک، دانشگاه یاسوج

چکیده

در این مقاله، با استفاده از تئوری الاستیسیته غیرمحلی اثر مقیاس کوچک بر روی ارتعاش آزاد غیرخطی نانو ورق نازک همگن مورد بررسی قرار گرفته است. فرمول بندی براساس تئوری ورق کلاسیک (مدل کیرشهف) انجام گرفته و مدل غیرخطی فون کارمن در روابط جابجایی کرنشی استفاده شده است. جهت در نظر گرفتن مقیاس کوچک و اثر غیر خطی هندسی، معادلات دیفرانسیل حاکم بر اساس تئوری الاستیسیته غیرمحلی به همراه مدل هندسی فون کارمن استخراج و از اثر اینرسی چرخشی و تنش برشی چشم پوشی شده است. با وجود مرتبه سه غیرخطی، معادله دافینگ به کمک انتگرال بیضوی جهت محاسبه فرکانسهای طبیعی حل گردیده است. همچنین بعضی پاسخهای تحلیلی جابجایی عمودی به کمک توابع بیضوی ژاکوبی ارائه شده است. راندمان و دقت روش با مقایسه نتایج بدست آمده با مقالات در دسترس مورد بررسی قرار گرفته است. اثرات پارامترهای مختلف بر روی ارتعاشات غیرخطی نانو ورق استخراج شده و همچنین احتمال وقوع پدیده تشدید داخلی در نانو ورق مستطیلی مورد بررسی قرار گرفته است.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Analytical Solution of Non-Linear Free Vibration of Thin Rectangular Plates with Various Boundary Conditions Based On Non-Local Theory

نویسندگان [English]

  • GholamAli Varzandian 1
  • Sima Ziaei 2
چکیده [English]

This article deals with the small-scale effect on the nonlinear free vibration of isotropic thin nano-plate using the nonlocal elasticity plate theory. The formulations are based on the Kirchhoff's plate theory, and von Karman-type nonlinearity is considered in strain displacement relations. To include the small scale and the geometrical nonlinearity effects, the governing differential equations are derived based on the nonlocal elasticity theory in conjunction with the von Karman geometrical model in which the effects of rotary inertia and transverse shear are neglected. With cubic non-linearities, Duffing's equation is solved by elliptic integral and natural frequencies are obtained. Also by means of Jacobi elliptic functions, some analytical solutions for deflection of plate are presented. The efficiency and accuracy of the method are demonstrated by comparing the developed result with those available in literature. The effects of various parameters on the nonlinear vibrations of nanoplates are presented. Occurrence probability of internal resonance in rectangular nanoplate is investigated.

کلیدواژه‌ها [English]

  • Nonlinear Free Vibration
  • Nonlocal Plate Theory
  • Elliptic Integral
  • Jacobi Elliptic Function
  • Internal Resonance

مراجع

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نشریه مهندسی مکانیک امیرکبیر
دوره 48، شماره 4
دی 1395
صفحه 331-346

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