مهندسی مکانیک مدرس

مهندسی مکانیک مدرس

رفتار آشوبناک نانو‌تیر بر روی بستر ویسکوالاستیک غیرخطی تحت تحریک هارمونیک با استفاده از تئوری غیرموضعی

نویسندگان
1 دانشجوی دکترای دانشگاه فردوسی مشهد
2 استاد دانشگاه فردوسی مشهد
چکیده
در این مقاله، ارتعاشات غیرخطی یک نانوتیر اویلر برنولی واقع بر بستر ویسکوالاستیک غیرخطی مورد بررسی قرار می‌گیرد. فرض می‌شود که نانو تیر در معرض یک نیروی هارمونیک قرار دارد که می تواند تخمینی از یک میدان الکتروستاتیک باشد. بستر ویسکوالاستیک غیرخطی برای دو حالت دارای سخت شوندگی و نرم شوندگی درنظر گرفته می‌شود. با توجه به مدل‌سازی در مقیاس نانو، معادلات دینامیک غیرخطی نانوتیر مورد نظر از روش تئوری الاستیسیته غیرموضعی ارینگن و با صرف‌نظر از اینرسی درون صفحه‌ای به‌دست می‌آید. با استفاده از روش گالرکین و شکل مود اول، معادله دیفرانسیل مشتقات پاره‌ای به‌دست آمده به معادله دیفرانسیل معمولی تبدیل می‌شود. پس از محاسبه نقاط تعادل سیستم و مشاهده دوشاخگی هیتروکلنیک، مدارهای هیتروکلنیک تعیین می‌شوند. سپس با استفاده از روش انتگرال ملنیکوف حرکت آشوبناک سیستم به‌صورت تحلیلی بررسی شده و محدوده امن رفتار سیستم با توجه به فضای پارامتری مساله مشخص می‌شود. نتایج نشان می‌دهد که وقتی بستر ویسکوالاستیک دارای خاصیت سخت شوندگی باشد، بروز رفتار آشوبناک در سیستم نمی‌تواند مورد انتظار باشد. مشاهده می‌شود که استفاده از تئوری الاستیسیته غیرموضعی برای بررسی رفتار آشوبناک نانو‌تیرها ضروری بوده و عدم استفاده از این تئوری نتایج متفاوتی می‌دهد و ممکن است سیستم را در ناحیه غیرامن قرار دهد.
کلیدواژه‌ها

عنوان مقاله English

Chaotic behavior of nonlocal nanobeam resting on a nonlinear viscoelastic foundation subjected to harmonic excitation

نویسندگان English

Mir Massoud 1
Masoud Tahani 2
1 Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
2 Ferdowsi University of Mashhad
چکیده English

In this paper, the nonlinear vibration of a Euler–Bernoulli nanobeam resting on a non-linear viscoelastic foundation is investigated. It is assumed that the nanobeam is subjected to a harmonic excitation that can be representative of an electrostatic field. The non-linear viscoelastic foundation is considered for both hardening and softening cases. By neglecting of the in-plane inertia, Eringen's nonlocal elasticity theory is used to model and derive the equation of motion of the nanobeam. Using the Galerkin method and the first mode shape, the obtained partial differential equation is reduced to the ordinary differential equation. Calculating the system's equilibrium points lead to heteroclinic bifurcation and the heteroclinic orbits are obtained. Then, using the Melnikov integral method, the chaotic motion of the system is studied analytically, and the safe region of the system is determined respect to the parametric space of the problem. When the viscoelastic foundation has a hardening characteristic, the chaotic behavior in the system does not occur. It has been observed that the use of nonlocal elasticity theory is necessary to investigate the chaotic behavior of nanobeam, and using the classical theory of elasticity may place the system in the chaotic region.

کلیدواژه‌ها English

Nanobeam
Nonlocal elasticity
Viscoelastic foundation
Chaos
Melnikov analysis
[1] X. Li, B. Bhushan, K. Takashima, C. W. Baek, Y. K. Kim, Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques, Ultramicroscopy, Vol. 97, No. 1–4, pp. 481-494, 2003.
[2] Y. Moser, M. A. M. Gijs, Miniaturized flexible temperature sensor, Journal of Microelectromechanical Systems, Vol. 16, No. 6, pp. 1349-1354, 2007.
[3] J. Pei, F. Tian, T. Thundat, Glucose biosensor based on the microcantilever, Analytical Chemistry, Vol. 76, No. 2, pp. 292-297, 2004.
[4] A. C. Eringen, Nonlocal polar elastic continua, International Journal of Engineering Science, Vol. 10, No. 1, pp. 1-16, 1972.
[5] A. C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, Vol. 54, No. 9, pp. 4703-4710, 1983.
[6] A. C. Eringen, Nonlocal Continuum Field Theories, First Edition, pp. 31-48, New York: Springer, 2002.
[7] J. N. Reddy, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science, Vol. 45, No. 2–8, pp. 288-307, 2007.
[8] M. Aydogdu, A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration, Physica E: Low-dimensional Systems and Nanostructures, Vol. 41, No. 9, pp. 1651-1655, 2009.
[9] J. N. Reddy, Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, International Journal of Engineering Science, Vol. 48, No. 11, pp. 1507-1518, 2010.
[10] A. GharehKhani, E. Abbaspoure-Sani, Study of static deflection and instability voltage of phase shifter micro-switches using a nonlinear beam model and non-localized elasticity theory, Modares Mechanical Engineering, Vol. 99, No. 9, pp. 9-99, 1396 .(in Persian فارسی(
[11] C. M. C. Roque, A. J. M. Ferreira, J. N. Reddy, Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method, International Journal of Engineering Science, Vol. 49, No. 9, pp. 976-984, 2011.
[12] I. Karimipöur, Y. Tadi Beni, A. Koochi, M.R. Abadyan, Using couple stress theory for modeling the size-dependent instability of double-sided beam-type nanoactuators in the presence of Casimir force, Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 38, No. 6, pp. 1779- 1795, 2015.
[13] Y. Tadi Beni, I. Karimipöur, M. R. Abadyan, Modeling the instability of electrostatic nano-bridges and nano-cantilevers using modified strain gradient theory, Applied Mathematical Modelling, Vol. 39, No. 9, pp. 2633– 2648, 2015.
[14] R. Ansari Khalkhali, A. Norouzzadeh, R. Gholami, Forced vibration analysis of conveying fluid carbon nanotube resting on elastic foundation based on modified couple stress theory, Modares Mechanical Engineering, Vol. 16, No. 8, pp. 27-34, 2015 .(in Persian فارسی(
[15] R. Ansari Khalkhali, A. Norouzzadeh, Nonlocal and surface effects on the buckling behavior of functionally graded nanoplates: An isogeometric analysis, Physica E: Low-dimensional Systems and Nanostructures, Vol. 84, pp. 84-97, 2016.
[16] R. Ansari, R. Gholami, S. Sahmani, A. Norouzzadeh, M. Bazdid-Vahdati, Dynamic stability analysis of embedded multi-walled carbon nanotubes in thermal environment, Acta Mechanica Solida Sinica, Vol. 28, No. 6, pp. 659-667, 2015.
[17] R. Ansari, A. Norouzzadeh, R. Gholami, M. Faghih Shojaei, M. Hosseinzadeh, Size-dependent nonlinear vibration and instability of embedded fluid-conveying SWBNNTs in thermal environment, Physica E: Low-dimensional Systems and Nanostructures, Vol. 61, pp. 148-157, 2014.
[18] H. T. Thai, A nonlocal beam theory for bending, buckling, and vibration of nanobeams, International Journal of Engineering Science, Vol. 52, pp. 56- 64, 2012.
[19] M. Şimşek, Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory, Composites Part B: Engineering, Vol. 56, pp. 621-628, 2014.
[20] Y. Z. Wang, F. M. Li, Nonlinear primary resonance of nano beam with axial initial load by nonlocal continuum theory, International Journal of NonLinear Mechanics, Vol. 61, pp. 74-79, 2014.
[21] F. Najar, S. El-Borgi, J. N. Reddy, K. Mrabet, Nonlinear nonlocal analysis of electrostatic nanoactuators, Composite Structures, Vol. 120, pp. 117-128, 2015.
[22] S. Hosseini-Hashemi, R. Nazemnezhad, H. Rokni, Nonlocal nonlinear free vibration of nanobeams with surface effects, European Journal of Mechanics - A/Solids, Vol. 52, pp. 44-53, 2015.
[23] S. El-Borgi, R. Fernandes, J. N. Reddy, Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation, International Journal of Non-Linear Mechanics, Vol. 77, pp. 348-363, 2015.
[24] A. R. Vosoughi, Nonlinear free vibration of functionally graded nanobeams on nonlinear elastic foundation, iranian journal of science and technology, Transactions of Civil Engineering, Vol. 40, No. 1, pp. 23-32, 2016.
[25] M. Soltanpour, M. Ghadiri, A. Yazdi, M. Safi, Free transverse vibration analysis of size dependent Timoshenko FG cracked nanobeams resting on elastic medium, Microsystem Technologies, pp. 1-18, 2016.
[26] N. Togun, Nonlocal beam theory for nonlinear vibrations of a nanobeam resting on elastic foundation, Boundary Value Problems, Vol. 2016, No. 1, pp. 1-14, 2016.
[27] K. Yagasaki, Bifurcations and chaos in vibrating microcantilevers of tapping mode atomic force microscopy, International Journal of Non-Linear Mechanics, Vol. 42, No. 4, pp. 658-672, 2007.
[28] H. S. Haghighi, A. H. D. Markazi, Chaos prediction and control in MEMS resonators, Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 10, pp. 3091-3099, 2010.
[29] M. S. Siewe, U. H. Hegazy, Homoclinic bifurcation and chaos control in MEMS resonators, Applied Mathematical Modelling, Vol. 35, No. 12, pp. 5533-5552, 2011.
[30] E. M. Miandoab, A. Yousefi-Koma, H. N. Pishkenari, F. Tajaddodianfar, Study of nonlinear dynamics and chaos in MEMS/NEMS resonators, Communications in Nonlinear Science and Numerical Simulation, Vol. 22, No. 1–3, pp. 611-622, 2015.
[31] E. Maani Miandoab, H. N. Pishkenari, A. Yousefi-Koma, F. Tajaddodianfar, Chaos prediction in MEMS-NEMS resonators, International Journal of Engineering Science, Vol. 82, pp. 74-83, 2014.
[32] J. Awrejcewicz, A. V. Krysko, V. Dobriyan, I. V. Papkova, V. A. Krysko, Chaotic and synchronized dynamics of non-linear Euler–Bernoulli beams, Computers & Structures, Vol. 155, pp. 85-96, 2015.
[33] F. Tajaddodianfar, H. Nejat Pishkenari, M. R. Hairi Yazdi, Prediction of chaos in electrostatically actuated arch micro-nano resonators: Analytical approach, Communications in Nonlinear Science and Numerical Simulation, Vol. 30, No. 1–3, pp. 182-195, 2016.
[34] M. Maleki, H. Nahvi, Nano-resonator dynamic behavior based on nonlocal elasticity theory, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 229, No. 14, pp. 2665-2671, October 1, 2015, 2015.
[35] D. Younesian, S. R. Marjani, E. Esmailzadeh, Nonlinear vibration analysis of harmonically excited cracked beams on viscoelastic foundations, Nonlinear Dynamics, Vol. 71, No. 1, pp. 109-120, 2013.
[36] H. Norouzi, D. Younesian, Chaotic vibrations of beams on nonlinear elastic foundations subjected to reciprocating loads, Mechanics Research Communications, Vol. 69, pp. 121-128, 2015.
[37] M. Şimşek, Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He’s variational method, Composite Structures, Vol. 112, pp. 264-272, 2014.
[38] A. Norouzzadeh, R. Ansari, Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity, Physica E: Low-dimensional Systems and Nanostructures, Vol. 88, pp. 194-200, 2017.
[39] A. Norouzzadeh, R. Ansari, H. Rouhi, Pre-buckling responses of Timoshenko nanobeams based on the integral and differential models of nonlocal elasticity: An isogeometric approach, Applied Physics A, Vol. 123:330, 2017.
[40] N. Challamel, C. M. Wang, The small length scale effect for a non-local cantilever beam: a paradox solved, Nanotechnology, Vol. 19, No. 34, pp. 34- 57, 2008.
[41] H. S. Shen, F. W. Williams, Postbuckling analysis of imperfect composite laminated plates on non-linear elastic foundations, International Journal of Non-Linear Mechanics, Vol. 30, No. 5, pp. 651-659, 1995/09/01, 1995.
[42] H. S. Shen, F. W. Williams, Thermomechanical postbuckling analysis of imperfect laminated plates of softening nonlinear elastic foundations, Composite Structures, Vol. 40, No. 1, pp. 55-66, 1997.
[43] H. S. Shen, Thermal postbuckling analysis of imperfect reissner-mindlin plates on softening nonlinear elastic foundations, Journal of Engineering Mathematics, Vol. 33, No. 3, pp. 259-270, 1998.
[44] S. A. Emam, A static and dynamic analysis of the postbackling of geometrically imperfect composite beams, Composite Structure, Vol. 90, No. 2, pp. 247-253, 2009.
[45] P. H. John Guckenheimer, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, First Edition, pp. 184-193, New York: Springer, 1983.
[46] S. Wiggins, Global Bifurcations and Chaos, pp. 426-429, New York: Springer, 1988.
[47] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, pp. 687-711, New York: Springer, 2003.
[48] F. C. Moon, Chaotic Vibrations: An Introduction for Applied Scientists and Engineers, pp. 191-200, John Wiley & Sons Inc., Hoboken, New Jersey, 2005.