نوع مقاله : مقاله پژوهشی

نویسندگان

1 دانشجوی کارشناسی ارشد، مهندسی مکانیک، دانشگاه خواج ه نصیرالدین طوسی، تهران.

2 استاد، مهندسی مکانیک، دانشگاه خواج ه نصیرالدین طوسی، تهران.

10.22068/jstc.2019.106866.1542

چکیده

در تحقیق حاضر، تحلیل ارتعاش آزاد غیرخطی ورق مستطیلی از جنس مواد مدرج تابعی با استفاده از نظریه تغییر شکل برشی مرتبه اول برای اولین بار مورد بررسی واقع شده‌است. برای این منظور ابتدا با استفاده از اصل هامیلتون، معادلات دیفرانسیل جزئی حرکت براساس تئوری تغییر شکل برشی مرتبه اول و با درنظرگرفتن روابط کرنش جابجایی غیرخطی ون‌کارمن استخراج شده‌اند. پس از آن با اعمال روش گالرکین، معادلات دیفرانسیل جزئی غیرخطی حرکت به معادلات دیفرانسیل معمولی غیرخطی تبدیل می‌شوند. سپس با استفاده از روش لیندشتد پوانکاره بهبودیافته، معادله غیرخطی حرکت عرضی ورق مدرج تابعی بصورت تحلیلی حل گشته و فرکانس های غیرخطی استخراج می‌شوند. خواص مواد مدرج تابعی ورق به صورت تابع توانی و پیوسته در راستای ضخامت فرض گردیده است. در نهایت، اثرات برخی پارامتر های کلیدی سیستم همچون دامنه ارتعاش بدون‌بعد، توان کسرحجمی ماده تابعی و همچنین نسبت ابعاد ورق روی نسبت فرکانس طبیعی غیرخطی به فرکانس طبیعی خطی مورد بحث قرار گرفته‌است. برای تایید و صحت فرمولاسیون مساله، نتایج این مطالعه با نتایج مقالات منتشر شده قبلی و همچنین راه حل عددی مقایسه گشته و تطابق خوبی دیده شده است.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Nonlinear free vibration analysis of functionally graded rectangular plate using modified Lindstedt-Poincare method

نویسندگان [English]

  • Soheil Hashemi 1
  • Ali -Asghar Jafari 2

1 MSc. Student, Department of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran.

2 Professor, Department of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran.

چکیده [English]

In this research, the nonlinear free vibration analysis of functionally graded (FG) rectangular plate is investigated analytically using first order shear deformation theory (FSDT) for the first time. For this purpose, firstly, using Hamilton principle, the partial differential equations of motion are developed based on first order shear deformation theory (FSDT) and von Karman nonlinearity strain displacement relations. Afterward, by applying Galerkin method, the nonlinear partial differential equations are transformed into nonlinear ordinary differential equations. Then, using the modified Lindstedt-Poincare method, the nonlinear equation of transverse motion of the FG plate is solved analytically to determine nonlinear frequency ratio. The material properties of the plate are assumed to be graded continuously according to power law distribution in the thickness direction. The effects of some key system parameters such as vibration amplitude, volume fraction index and aspect ratio on the nonlinear natural frequency ratio to linear natural frequency are discussed. To validate the analysis, the results of this study are compared with the results of previously published papers and numerical solution and good agreement has been observed.

کلیدواژه‌ها [English]

  • Nonlinear vibrations
  • Rectangular plate
  • Functionally graded materials
  • First order shear deformation theory
  • modified Lindstedt-Poincare method
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