حل دقیق برای معادلات فرکانسی ارتعاشات آزاد شعاعی و عرضی یک ورق دایره‌ای با شرایط مرزی مختلف

نوع مقاله : پژوهشی

نویسندگان

1 دانشگاه اصفهان

2 دانشگاه کاشان

چکیده

در این تحقیق، معادله‌های فرکانسی ارتعاشات آزاد شعاعی و عرضی یک ورق دایره‌ای با سه نوع تکیه گاه آزاد، ساده و گیردار به صورت حل دقیق بسته استخراج شده است. برای به دست آوردن معادلات حاکم بر ارتعاشات شعاعی و عرضی ورق دایره‌ای از اصل همیلتون استفاده شده است. در حالت ارتعاشات شعاعی، دو معادله دیفرانسیل وابسته به هم حاصل می‌شود که با استفاده از روش تجزیه هلمهولتز، آن دو معادله دیفرانسیل از یکدیگر مستقل می‌شوند و سپس با روش جداسازی متغیرها به صورت تحلیلی قابل حل خواهند بود. در حالت ارتعاشات عرضی ورق دایره‌ای، معادلات حاکم نیز با استفاده از روش جداسازی متغیرها حل شده و یک رابطه دقیق بسته برای معادله فرکانسی در هر شرط مرزی به دست می‌آید. در نهایت فرکانس‌های طبیعی حاصل از معادله‌های فرکانسی به دست آمده با نتایج المان محدود مقایسه گردیده است.

کلیدواژه‌ها


عنوان مقاله [English]

Exact solution for frequency equations of radial and transverse vibration of a circular plate with various boundary conditions

نویسندگان [English]

  • Mohammad Heidari-Rarani 1
  • Shahram Hosseini-Chaleshtori 2
  • Keivan Torabi 2
1
2
چکیده [English]

In this study, closed-form relations are obtained for the frequency equations of radial (in-plane) and transverse (out-of–plane) free vibration of a circular plate with free, simply-support, and clamped boundary conditions. Governing equations are obtained using Hamilton's principal. In the case of radial vibration, governing equations are coupled to each other. So, they are decoupled using Helmholtz decomposition technique and solved using separation of variables method. In the case of transverse vibration, governing equations are analytically solved using separation of variables method. Finally, natural frequencies obtained from the frequency equations are compared with finite element results for an isotropic homogeneous circular plate with various boundary conditions.

کلیدواژه‌ها [English]

  • Radial vibration
  • Transverse vibration
  • Circular plate
  • Exact solution
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