شیوه‌ای نوین در تعیین مدهای طبیعی و فرکانس‌های متناظر سیستم‌های با چند درجه آزادی بر اساس مفهوم گره ارتعاش

نوع مقاله : مقاله پژوهشی

نویسنده

مدرس دانشگاه/ دانشکده فنی و مهندسی دانشگاه گلستان

چکیده

روش‌­های گوناگون تحلیل دینامیکی سازه­‌ها با مفهوم مدهای طبیعی سر و کار دارند و در ریاضیات به مدهای طبیعی اصطلاحا بردارهای مشخصه گفته می­‌شود. روش مرسوم برای تعیین مدهای طبیعی یک سیستم حل مسئله مقادیر ویژه است. در این روش بر اساس ماتریس­‌های سختی و جرم، ماتریس مشخصه ساخته می­‌شود که با صفر قرار دادن دترمینان آن، فرکانس­2های طبیعی سیستم به ­دست می­‌آید. با تعیین فرکانس­‌ها، بردار مدهای ارتعاش ساخته می­‌شوند. روشی که در این مقاله بیان می‌­شود مبتنی بر تعیین مکان (مکان­هایی) است که در هر شکل مد، جابجایی آن (آن­ها) صفر است که به گره موسوم است. در واقع با این روش، یک سیستم با چند درجه آزادی در هر مد به چند سیستم با یک درجه آزادی تبدیل می­‌شود که در محل گره­‌های فرضی از یکدیگر جدا شده‌­اند. جرم‌­های سیستم­‌های مزبور به­ طور هم­زمان، با یک فرکانس ارتعاش می­‌کنند که همان فرکانس ارتعاش در مد مورد نظر است. هر فنری که بین دو جرم مجاور قرار دارد، در محل گره به دو فنر متوالی تفکیک می­‌شود، به ­طوری که فنر اولیه دارای سختی معادل فنرهای متوالی (سری) است. این روش درک فیزیکی مناسبی برای مفهوم مد ارتعاش ارائه می­‌کند. روش به ­کار رفته دقیق بوده و در برخی موارد، ساده و سریع­تر از روش مرسوم عمل می­‌کند.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

A new method for determining natural modes and their frequencies with the concept of node in vibrations of M-DOFs

نویسنده [English]

  • Aliasghar Safavi
Lecturer at Engineering Department of Golestan University
چکیده [English]

This paper evaluates the vibration of M-DOF systems by calculating the natural frequencies and mode shapes. The introduced method is established on the base of the node concept, which is the point of a mode shape with zero displacement. In this method, a system with two or more degrees of freedom is transformed into two or more isolated systems with one- DOF. Those systems are isolated in node places and vibrate with the same frequencies in every mode. Each spring located between two adjacent lumped masses will be converted to series combination of two separated springs. The stiffness of the first spring is equal to the effective stiffness of the two series separated springs. The proposed method provides a good physical understanding about the concept of vibration modes. Besides, this method is accurate and sometimes is simpler and quicker than the common method.

کلیدواژه‌ها [English]

  • Natural modes
  • Natural Frequencies
  • node
  • M-DOFs system
  • one- DOF systems
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