مهندسی مکانیک مدرس

مهندسی مکانیک مدرس

وابستگی ابعادی ارتعاشات غیرخطی میکروتیر با مقطع غیریکنواخت و شرایط مرزی مختلف

نویسندگان
عطیه اندخشیده 1
ستار مالکی 2
حسین کارامد 3
1 استادیار- گروه مهندسی مکانیک-دانشکده مهندسی-دانشگاه صنعتی قوچان-قوچان-ایران
2 عضو هیات علمی گروه مکانیک/دانشگاه صنعتی قوچان
3 دانشگاه صنعتی قوچان
چکیده
در این پژوهش، برای اولین بار، تاثیر غیریکنواختی مقطع میکروتیر و شرایط مرزی مختلف بر ارتعاشات غیرخطی میکروتیر با در نظر گرفتن رفتار وابسته به ابعاد براساس تئوری‌ تنش کوپل اصلاح شده، بررسی می‌شود. با استفاده از اصل همیلتون، معادله حاکم بر میکروتیر اویلر- برنولی، با در نظر گرفتن غیرخطی هندسی فون کارمن و بر اساس تئوری‌ تنش کوپل اصلاح شده استخراج می‌گردد. سپس معادله ارتعاشات غیرخطی حاصل، با استفاده از روش مشتقات تربیعی بهبود یافته و روش تکرار مستقیم حل می‌شود تا فرکانس طبیعی غیرخطی بدست آید. در این مرحله، با استفاده از روش گلرکین، معادلات دیفرانسیل جزئی غیرخطی حاکم بر ارتعاشات به معادله دیفرانسیل معمولی غیرخطی از نوع دافینگ با متغیر زمان کاهش می‌یابد. معادله حاصل، با استفاده از ماتریس دیفرانسیل گیر اسپکترال که بر اساس مشتقات توابع پریودیک پایه تعریف شده است، تفکیک می‌گردد. سپس با استفاده از روش طول کمان، معادله غیرخطی پارامتری حاصل، حل می‌شود و پاسخ فرکانسی ارتعاشات اجباری غیرخطی میکروتیر بدست می‌آید. نهایتا فرکانس طبیعی غیرخطی و پاسخ فرکانسی میکروتیر برای مقاطع غیریکنواخت و شرایط مرزی مختلف محاسبه و تحلیل می‌شود. نتایج پژوهش جاری نشان میدهد که ارتعاشات غیرخطی آزاد و اجباری میکروتیر وابسته به ابعاد است. همچنین برای تیرغیریکنواخت، این وابستگی ابعادی، چشمگیرتر و برای شرایط مرزی مختلف، متفاوت است. راستی‌آزمایی روش با مقایسه نتایج با نتایج حل دقیق در حالت ساده شامل مقطع یکنواخت و تکیه‌گاه ساده صورت گرفته که بیانگر دقت خوب روش حاضر می‌باشد.
کلیدواژه‌ها
وابستگی ابعادی
ارتعاشات غیرخطی
پاسخ فرکانسی
میکروتیر
تئوری‌ تنش کوپل اصلاح شده

موضوعات

عنوان مقاله English

Size-dependent nonlinear vibration of non-uniform microbeam with various boundary conditions

نویسندگان English

Atieh Andakhshideh 1
Sattar Maleki 2
Hossien Karamad 3
1 Department of Mechanical Engineering, Quchan University of technology, Quchan, Iran
2 Quchan University of Technology
3 Quchan University of Technology
چکیده English

In this article, for the first time, the effect of non-uniformity of microbeam cross section and various boundary conditions on the nonlinear vibration of microbeam is investigated considering the size dependent behavior based on modified couple stress theory. Using the Hamilton’s principle, the governing equation of Euler–Bernoulli microbeam with von Karman geometric nonlinearity based on the modified couple stress theory is derived. The nonlinear vibration governing equation is then solved using the Generalized Differential Quadrature method (GDQ) and direct iterative method to obtain the nonlinear natural frequencies. In this step, the Galerkin method is used to reduce the nonlinear PDE governing the vibration into a time-dependent ODE of Duffing-type. The time domain is then discretized via spectral differentiation matrix operators which are defined based on the derivatives of a periodic base function. Next, the nonlinear parametric equation is solved using pseudo arc-length method and the frequency–response curves of microbeam nonlinear forced vibration is obtained. Finally, nonlinear natural frequency and frequency response of microbeam with various non-uniformity of cross sections and boundary conditions are obtained. Present results show that, the nonlinear free and forced vibration of microbeam is size dependent. Moreover, this size dependency is more significant for non-uniform microbeam and is deferent for various boundary conditions. The result of present method for simple case including uniform section and simply supported boundary condition is validated with that of exact method and have good agreement.

کلیدواژه‌ها English

Size dependent
Nonlinear Vibration
Frequency Response
Microbeam
Modified couple stress theory
[1] L. J. Currano, M. Yu, B. Balachandran, Latching in a MEMS shock sensor: Modeling and experiments, Sensors and Actuators A: Physical, Vol. 159, No. 1, pp. 41-50, 2010.
[2] S. Chaterjee, G. Pohit, A large deflection model for the pull-in analysis of electrostatically actuated microcantilever beams, Journal of sound and vibration, Vol. 322, No. 4, pp. 969-986, 2009.
[3] M. H. Mahdavi, A. Farshidianfar, M. Tahani, S. Mahdavi, H. Dalir, A more comprehensive modeling of atomic force microscope cantilever, Ultramicroscopy, Vol. 109, No. 1, pp. 54-60, 2008.
[4] M. A. Mohammadi, K. A. Yousefi, M. E. Maani, M. Karipour, Dynamics behavior analysis of atomic force microscope based on gradient theory, Modares Mechanical Engineering, Vol. 16, No. 9, pp. 155-164, 2016. (in Persian فارسی )
[5] M. Abbasi, Nonlinear vibration analysis of a dynamic atomic force microscope microcantilever in the tapping mode based on the modified couple stress theory, Modares Mechanical Engineering, Vol. 14, No. 11, pp. 9-17, 2015. (in Persian فارسی )
[6] E. M. Abdel-Rahman, M. I. Younis, A. H. Nayfeh, Characterization of the mechanical behavior of an electrically actuated microbeam, Journal of Micromechanics and Microengineering, Vol. 12, No. 6, pp. 759, 2002.
[7] Y. Ahn, H. Guckel, J. D. Zook, Capacitive microbeam resonator design, Journal of Micromechanics and Microengineering, Vol. 11, No. 1, pp. 70, 2001.
[8] L. D. Gabbay, J. E. Mehner, S. D. Senturia, Computer-aided generation of nonlinear reduced-order dynamic macromodels. I. Non-stress-stiffened case, Journal of Microelectromechanical Systems, Vol. 9, No. 2, pp. 262-269, 2000.
[9] A. H. Nayfeh, M. I. Younis, E. M. Abdel-Rahman, Dynamic pull-in phenomenon in MEMS resonators, Nonlinear dynamics, Vol. 48, No. 1-2, pp. 153-163, 2007.
[10] S. Govindjee, J. L. Sackman, On the use of continuum mechanics to estimate the properties of nanotubes, Solid State Communications, Vol. 110, No. 4, pp. 227-230, 1999.
[11] Q. Ma, D. R. Clarke, Size dependent hardness of silver single crystals, Journal of Materials Research, Vol. 10, No. 4, pp. 853-863, 1995.
[12] J. Stölken, A. Evans, A microbend test method for measuring the plasticity length scale, Acta Materialia, Vol. 46, No. 14, pp. 5109-5115, 1998.
[13] A. C. Chong, D. C. Lam, Strain gradient plasticity effect in indentation hardness of polymers, Journal of Materials Research, Vol. 14, No. 10, pp. 4103-4110, 1999.
[14] E. Cosserat, F. Cosserat, Deformable Bodies, Scientific Library A. Hermann and Sons, 1909.
[15] R. D. Mindlin, Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, Vol. 16, No. 1, pp. 51-78, 1964.
[16] R. Mindlin, H. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and analysis, Vol. 11, No. 1, pp. 415-448, 1962.
[17] R. D. Mindlin, Second gradient of strain and surface-tension in linear elasticity, International Journal of Solids and Structures, Vol. 1, No. 4, pp. 417-438, 1965.
[18] R. Mindlin, N. Eshel, On first strain-gradient theories in linear elasticity, International Journal of Solids and Structures, Vol. 4, No. 1, pp. 109-124, 1968.
[19] W. KOlTER, Couple stresses in the theory of elasticity, Proc. Koninklijke Nederl. Akaad. van Wetensch, Vol. 67, 1964.
[20] F. Yang, A. Chong, D. C. C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, Vol. 39, No. 10, pp. 2731-2743, 2002.
[21] A. R. Hadjesfandiari, G. F. Dargush, Couple stress theory for solids, International Journal of Solids and Structures, Vol. 48, No. 18, pp. 2496-2510, 2011.
[22] S. Park, X. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, Vol. 16, No. 11, pp. 2355, 2006.
[23] S. Kong, S. Zhou, Z. Nie, K. Wang, The size-dependent natural frequency of Bernoulli–Euler micro-beams, International Journal of Engineering Science, Vol. 46, No. 5, pp. 427-437, 2008.
[24] W. Xia, L. Wang, L. Yin, Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration, International Journal of Engineering Science, Vol. 48, No. 12, pp. 2044-2053, 2010.
[25] M. H. Kahrobaiyan, M. Asghari, M. Hoore, M. T. Ahmadian, Nonlinear size-dependent forced vibrational behavior of microbeams based on a non-classical continuum theory, Journal of Vibration and Control, Vol. 18, No. 5, pp. 696-711, 2012.
[26] D. C. Lam, F. Yang, A. Chong, J. Wang, P. Tong, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, Vol. 51, No. 8, pp. 1477-1508, 2003.
[27] S. Kong, S. Zhou, Z. Nie, K. Wang, Static and dynamic analysis of micro beams based on strain gradient elasticity theory, International Journal of Engineering Science, Vol. 47, No. 4, pp. 487-498, 2009.
[28] B. Wang, J. Zhao, S. Zhou, A micro scale Timoshenko beam model based on strain gradient elasticity theory, European Journal of Mechanics-A/Solids, Vol. 29, No. 4, pp. 591-599, 2010.
[29] M. Kahrobaiyan, M. Asghari, M. Rahaeifard, M. Ahmadian, A nonlinear strain gradient beam formulation, International Journal of Engineering Science, Vol. 49, No. 11, pp. 1256-1267, 2011.
[30] L. N. Trefethen, Spectral methods in MATLAB: SIAM, 2000.
[31] R. Ansari, V. Mohammadi, M. F. Shojaei, R. Gholami, S. Sahmani, On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory, Composites Part B: Engineering, Vol. 60, pp. 158-166, 2014.
[32] R. Bellman, B. Kashef, J. Casti, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Journal of computational physics, Vol. 10, No. 1, pp. 40-52, 1972.
[33] X. Wang, C. Bert, A. Striz, Differential quadrature analysis of deflection, buckling, and free vibration of beams and rectangular plates, Computers & structures, Vol. 48, No. 3, pp. 473-479, 1993.
مهندسی مکانیک مدرس
دوره 18، شماره 9
دی 1397
صفحه 189-198