تاثیر بستر الاستیک وینکلر بر ارتعاش آزاد تیر ماهیچه‌ای براساس تئوری الاستیسیته غیرمحلی

سلطانی, معصومه; محمدی, معصومه; عسگریان, بهروز

doi:10.22065/jsce.2019.176357.1807

تاثیر بستر الاستیک وینکلر بر ارتعاش آزاد تیر ماهیچه‌ای براساس تئوری الاستیسیته غیرمحلی

نوع مقاله : علمی - پژوهشی

نویسندگان

¹ استادیار گروه مهندسی عمران، دانشکده مهندسی، دانشگاه کاشان ،کاشان، ایران

² دانشجوی کارشناسی ارشد مهندسی سازه، گروه مهندسی عمران، دانشکده مهندسی، دانشگاه کاشان، کاشان، ایران

³ استاد دانشکده مهندسی عمران، دانشگاه صنعتی خواجه نصیرالدین طوسی، تهران، ایران

10.22065/jsce.2019.176357.1807

چکیده

در مقاله پیش‌رو، ارتعاش آزاد تیر با مقطع متغیر متکی بر بستر الاستیک وینکلر و با در نظر گرفتن تئوری الاستیسیته غیرمحلی مورد مطالعه قرار گرفته است. روند کار به این صورت است که در مرحله نخست، معادله حرکت براساس فرضیه تغییر شکل‌های کوچک و مدل الاستیسیته غیرمحلی ارینگن به کمک اصل همیلتون به‌دست می‌آید. در ادامه، از روش نیمه تحلیلی بسط سری‌های توانی برای حل معادله دیفرانسیل ارتعاش آزاد عضو غیرمنشوری متکی بر بستر الاستیک استفاده شده است و با جایگذاری شرایط مرزی (دو شرط برای هر انتهای عضو) و حل مسئله مقادیر ویژه، مقدار فرکانس طبیعی ارتعاش و همچنین فرم تغییر شکل عضو مدنظر تحت آنالیز ارتعاش آزاد نامیرا تعیین می‌گردد. در انتها نتایج حاصل از این تحقیق در خصوص تیر همگن با مقطع غیریکنواخت با نتایج دیگر تحقیقات جهت بررسی صحت و دقت محاسبات مقایسه گردیده و تاثیر پارامترهای مختلفی همچون پارامتر ارینگن، ضریب باریک شوندگی مقطع و بستر الاستیک وینکلر روی ارتعاش آزاد بدون بعد تیر با شرایط مرزی مختلف به تفصیل بررسی شده است. نتایج بدست آمده از این پژوهش نشان می‌دهند که با افزایش پارامتر ارینگن و باریک شدگی نیم‌رخ عضو مقدار فرکانس طبیعی بدون بعد کاهش می‌یابد، در حالی که با افزایش ثابت فنری نوع وینکلر در بازه استفاده شده در این مقاله فرکانس ارتعاشی به شدت افزایش می‌یابد.

کلیدواژه‌ها

عنوان مقاله [English]

Effect of Winkler Elastic Foundation on Free Vibration of Tapered Beam Based on Non-Local Elasticity Theory

نویسندگان [English]

Masoumeh Soltani ¹
Masooma Mohammadi ²
Behrouz Asgarian ³

¹ Assistant Professor, Department of civil engineering, University of Kashan, Kashan, Iran

² MSc student in structural engineering, Department of civil engineering, University of Kashan, Kashan, Iran

³ Professor, Faculty of civil engineering, K. N. Toosi University of Technology, Tehran, Iran

چکیده [English]

In this paper, the influence of Winkler type elastic foundation on the free vibration behavior of non-local Euler-Bernoulli beam with varying cross-section is semi-analytically investigated. For this purpose, the governing equation of motion is derived based on nonlocal elasticity Eringen’s model and Hamilton’s principle. The power series approximation is applied to solve the fourth order differential equation, since in the presence of variable cross-section, stiffness quantities are not constant. Based on this semi-analytical methodology, displacement component and cross-section properties should be expanded in terms of power series of a known degree. The natural frequencies of non-local beam with variable cross-section are then derived by imposing the boundary conditions and solving the eigenvalue problem. The results of this research are compared with the obtained results by other researchers and there is a good agreement. At the end, the effects of different parameters such as: end conditions, nonlocal Eringen’s parameter, tapering ratio and Winkler spring constant on non-dimensional natural frequency of nano-beam are studied in detail. The outcomes of this study indicate that the increase of nonlocal parameter and non-uniformity ratio causes to decrease the dimensionless natural frequency. However, with increasing the Winkler elastic constant, the non-local frequencies increase. Furthermore, it is shown that the effect of Winkler elastic foundation is predominate than the tapering ratio and Eringen’s parameter on the natural frequency of nano-beams.

کلیدواژه‌ها [English]

Natural frequency Non
local elasticity theory Tapered beam Elastic foundation Power series method

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