مهندسی مکانیک مدرس

مهندسی مکانیک مدرس

بررسی پدیده تشدید درونی و انتقال انرژی بین مودهای ارتعاشی در تیر ترک‌دار

نوع مقاله : پژوهشی اصیل

نویسندگان
گروه مهندسی مکانیک، دانشکده مهندسی مکانیک، دانشگاه تبریز، تبریز، ایران
چکیده
در این مقاله معادلات حرکت غیرخطی تیر یک‌سر گیر‌دار و یک‌سر مفصل با ترک‌باز استخراج‌شده و با حل آن به مطالعه پدیده تشدید درونی در تیر ترک‌دار پرداخته شده است. ترک به صورت یک فنر پیچشی مدل‌سازی شده و تیر ترک‌دار به صورت دو تیر مجزا که با یک فنر پیچشی به هم متصل شده‌اند در نظر گرفته شده است. معادله حرکت تیر ترک‌دار با فرض غیرخطی هندسی استخراج و با استفاده از روش گالرکین به مجموعه‌ای از معادلات غیرخطی برای مودهای ارتعاشی تبدیل شدند و با استفاده از روش اغتشاشات حل شدند. با توجه به وابستگی انرژی مکانیکی تیر با دامنه نوسان آن، برای بررسی انتقال انرژی بین مودهای ارتعاشی و تاثیر ترک در آن، دامنه آنی مودهای ارتعاشی به دست آمد. نتایج به‌دست‌آمده نشان می‌دهد که در تیر ترک‌دار مقدار انرژی انتقالی بین مودها کمتر از تیر سالم بوده ولی نرخ تکرار انتقال انرژی بیشتر از تیر سالم است و نرخ جا‌به‌جایی انرژی بین مودها با افزایش عمق ترک با شیب تندتری افزایش می‌یابد. همچنین پاسخ ارتعاشی به‌دست‌آمده برای تیر ترک‌دار با میرائی‌های مختلف نشان می‌دهد که مقدار و سرعت انتقال انرژی وابستگی ناچیزی به میرائی سیستم دارد. برای صحه‌سنجی نتایج به‌دست‌آمده از روش اغتشاشات، معادلات حرکت با استفاده از روش عددی حل شده و نتایج به‌دست‌آمده حاکی از همخوانی کامل نتایج به‌دست‌آمده از روش‌های تحلیلی و عددی است.
کلیدواژه‌ها

موضوعات


عنوان مقاله English

Investigating the internal resonance and energy exchange between the vibration modes of a cracked beam

نویسندگان English

M. Rezaee
V. Shaterian_Alghalandis
Department of Mechanical Engineering, Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran
چکیده English

The equations of nonlinear motion of clamped-hinged beam with an open crack were extracted and through solving them, the internal resonance in the cracked beam was studied. To this end, the crack was modeled as a torsional spring and the cracked beam was considered as two beam segments connected by a torsional spring. The equations of motion of the cracked beam were extracted considering the geometrical nonlinearity. Then, using the Galerkin’s method, these equations were changed to a set of nonlinear differential equations for vibration modes which were solved by the perturbation method. Since the mechanical energy of the beam in each mode depends on the instantaneous amplitude of vibration of the beam at the corresponding mode, so to analyze the influence of the crack on the energy exchange between the modes, the instantaneous amplitudes of the vibration modes were obtained. The results show that in the cracked beam the magnitude of the energy exchanged between the modes is less and the frequency is more than that in the intact beam. Also, by increasing the crack depth the frequency of energy exchange between the modes increases. The Vibration response obtained for the cracked beam with various amounts of the damping ratios shows that the frequency and the amplitude of energy exchange between the modes are independent of the system damping. To validate the results by the perturbation method, the equations of motions are also solved by a numerical method and the obtained results are in agreement with the results of the analytical method.

کلیدواژه‌ها English

Clamped-Hinged Beam
Open Crack
Geometrical Nonlinearity
Internal resonance
Hilbert transform
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