تحلیل تشدید‌های مافوق هارمونیک و مادون هارمونیک یک غشای مستطیلی هایپرالاستیک بر بستر الاستیک غیرخطی با استفاده از روش مقیاس‌های چندگانه

نوع مقاله : مقاله پژوهشی

نویسندگان

دانشگاه صنعتی شاهرود

چکیده

در این تحقیق ارتعاشات غیرخطی غشای مستطیلی هایپرالاستیک بر بستر الاستیکغیرخطی وینکلر-پسترناک تحت فشار خارجی گسترده و یکنواخت مورد بررسی قرارگرفته است. ساختار ماده‌ای این غشا تراکم‌ناپذیر، همگن و ایزوتروپیک می‌باشد. بستر الاستیک غیرخطی شامل دو جمله خطی وینکلر و پسترناک و یک جمله غیرخطی درجه سه وینکلر می‌باشد. با استفاده از نظریه غشا، اصل هامیلتون و با فرض تغییرشکل محدود، معادلات حاکم بر سیستم استخراج گردیده است. هم‌چنین با توجه به تابع چگالی انرژی کرنشی مدل ساختاریهایپرالاستیک نئو-هوکین،انرژی جنبشی، کار ناشی از بارگذاری گسترده و فشار یکنواخت و اثرات میرایی تعیین می‌گردند. با اعمال روش گلرکین، معادله حرکت غیرخطی با مشتقات جزئی درجهت عرضی به معادلات دیفرانسیل معمولی تبدیل می‌شود.سپس با استفاده از روش مقیاس‌‌های چندگانه به تحلیل تشدید‌های مافوق و مادون هارمونیک شامل تشدید‌های مافوق هارمونیک 3:1 و مادون هارمونیک 1:3، مافوق هارمونیک 5:1 و مادون هارمونیک 1:5، مافوق هارمونیک 7:1 و مادون هارمونیک 1:7، پرداخته شده است. هم‌چنین نتایج حاصل از روش تحلیلی با نتایج موجود در ادبیات موضوع، مقایسه شده است. درانتها اثر پارامترهای مختلف سختی وینکلر، پسترناک، خواص مواد و هندسی مختلف بر روی رفتار ارتعاشی غشای مستطیلی هایپرالاستیک در تشدید‌های مختلف مافوق و مادون هارمونیک مورد بررسی قرار گرفته است.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Superharmonic and Subharmonic Resonance Analysis of A Rectangular Hyperelastic Membrane Resting on Nonlinear Elastic Foundation Using The Method of Multiple Scales

نویسندگان [English]

  • Sina Karimi
  • Habib Ahmadi
  • Kamran Foroutan
Shahrood University of Technology
چکیده [English]

In this paper, the nonlinear vibrations of a rectangular hyperelastic membrane resting on a nonlinear elastic Winkler-Pasternak foundation subjected to uniformly distributed hydrostatic pressure are investigated. The membrane is composed of an incompressible, homogeneous, and isotropic material. The elastic foundation includes two Winkler and Pasternak linear terms and a Winkler term with cubic nonlinearity. Using the theory of thin hyperelastic membrane, Hamilton’s principle, and assuming the finite deformations, the governing equations are obtained. Also, the kinetic energy, the work of uniform distributed force and pressure, and the effects of damping are determined, according to the strain energy function for neo-Hookean hyperelastic constitutive law. By applying Galerkin’s method, the nonlinear partial differential equation of motion in the transversal direction is transformed to the ordinary differential equations. Then, utilizing the method of multiple scales, the superharmonic and subharmonic resonances including the 1:3 superharmonic and 3:1 subharmonic, 1:5 superharmonic, and 5:1 subharmonic, 1:7 superharmonic, and 7:1 subharmonic are analyzed. Also, the analytical results are compared with those presented by other researchers. Finally, the effect of the Winkler and Pasternak stiffness, the material properties, and various geometrical characteristics on the superharmonic and subharmonic resonances of the vibration behavior of a rectangular hyperelastic membrane is investigated.

کلیدواژه‌ها [English]

  • Rectangular hyperelastic membrane
  • Nonlinear Winkler-Pasternak foundation
  • Superharmonic resonance
  • Subharmonic resonance
  • Multiple scales method

مراجع

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نشریه مهندسی مکانیک امیرکبیر
دوره 53، شماره 8
آبان 1400
صفحه 4535-4564

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