ارتعاشات طولی آزاد غیرخطی میله تحت کرنش محدود

نوع مقاله : مقاله پژوهشی

نویسندگان

دانشکده مهندسی مکانیک، دانشگاه یزد، یزد، ایران

چکیده

میله‌ها یکی از اعضای مهم در سازه‌های مهندسی هستند و تحلیل ارتعاشات میله به علت کاربرد وسیع آن در مهندسی دارای اهمیت زیادی است. بنابراین، درک چگونگی ارتعاشات غیرخطی محوری میله در شرایط تکیه‌گاهی متفاوت، با دامنه اولیه زیاد، بسیار مفید است. لذا در این مقاله، به تحلیل ارتعاشات میله با شرایط تکیه‌گاهی متفاوت، براساس کرنش محدود، بدون ساده‌سازی در روابط کرنش-جابجایی پرداخته شده و برای بدست آوردن معادله‌های حاکم از کرنش گرین- لاگرانژی، میرایی ساختاری و اصل هامیلتون استفاده شده است. سپس با استفاده از روش گالرکین معادله غیرخطی پارهای به معادله غیرخطی معمولی تبدیل شده است. معادله‌های حاصل بر خلاف اکثر مقالات که برای معادله ارتعاشاتی غیرخطی فقط جمله غیرخطی درجه سه را درنظر می‌گیرند، دارای جمله‌های غیرخطی درجه دو و سه هستند. این معادله‌ها به روش مقیاس‌های چندگانه حل شده و پاسخ ارتعاشاتی میله در دو حالت بدون میرایی و با میرایی با شرایط مختلف تکیه-گاهی بدست آمده است. برای بررسی دقت روش و صحت سنجی آن، نتایج به دست آمده از روش حاضر با نتایج روش عددی رانگ-کوتای درجه چهارم مقایسه گردیده که نشان می‌دهد روش حاضر دارای دقت مناسبی است. در پایان تحلیل حساسیت برای بررسی تأثیر ضریب‌های غیرخطی بر پاسخ ارتعاشاتی میله انجام شده است.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Nonlinear Longitudinal Free Vibration of a Rod Undergoing Finite Strain

نویسندگان [English]

  • B. Soleimani Roody
  • A. Fotouhi
  • M. Jalili
Department of Mechanical Engineering, Yazd University, Yazd, Iran.
چکیده [English]

Rods are one of significant engineering’s structures and vibration analysis of a rod because of extended application of it in engineering is very important. Therefore, understanding of longitudinal nonlinear vibration of rod with different boundary conditions and large amplitude is very useful. In this paper, vibration of a rod with different boundary conditions undergoing finite strain, without simplification in strain-displacement relations, is investigated. For obtaining governing equation, Green-Lagrange strain, structural damping and Hamilton principle are used and then Galerkin method is employed to convert nonlinear partial differential equation to nonlinear ordinary differential equation. In spite of many papers that only use of cubic term for nonlinearity, the governing equation has quadratic and cubic terms. The equations with and without damping, are solved with multiple time scales method. In order to verify the accuracy of this method, the results are compared with results of Runge-Kutta numerical method, which have good accuracy. Finally sensitivity analysis for understanding of influence of nonlinear coefficients on rod vibration answer is done.

کلیدواژه‌ها [English]

  • nonlinear vibration
  • Finite strain
  • Green-Lagrange strain
  • Multiple scales method
  • Sensitivity analysis
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