بررسی ارتعاشات آزاد یک نانو تیر مدرج هدفمند طولی به همراه جرم متمرکز با استفاده از روش عددی جدید توسعه تقریبی و تئوری غیرموضعی گرادیان کرنشی

نوع مقاله : مقاله پژوهشی

نویسندگان

1 مکانیک، دانشگاه زنجان، زنجان، ایران

2 گروه صنایع، مکانیک و هوافضا، دانشگاه فنی و مهندسی بوئین زهرا، بوئین زهرا، قزوین، ایران

3 دانشیار گروه مهندسی مکانیک معاون پژوهشی دانشکده مهندسی دانشگاه زنجان

چکیده

در این مقاله هدف، مطالعه ارتعاشات آزاد طولی یک نانو تیر مدرج هدفمند به همراه جرم متمرکز براساس تئوری غیرموضعی گرادیان کرنشی با استفاده از روش عددی توسعه تقریبی می‌باشد. امروزه با توجه به اهمیت و کاربرد روز افزون نانو سازه‌ها، بررسی و شناخت خصوصیات مکانیکی و ارتعاشی آن‌ها ضروری به نظر می‌رسد. به دلیل ابعاد کوچک سازه و رفتار وابسته به اندازه آن و عدم توانایی تئوری‌های کلاسیک در پیش‌بینی رفتارهای مکانیکی وابسته به اندازه، از تئوری‌های غیرکلاسیک استفاده گردیده است. معادلات حاکم و روابط شرایط مرزی با استفاده از اصل همیلتون استخراج گردیده و برای نانو تیر مدرج هدفمند با پنج شرط مرزی لولا-لولا، گیردار-آزاد، گیردار-گیردار، گیردار-لولا و گیردار-جرم متمرکز ارائه شده است. سپس، معادله دیفرانسیل به صورت تحلیلی حل گردیده است تا معادله‌های فرکانس برای پنج شرط مرزی به دست آیند. در ادامه با استفاده از روش عددی توسعه تقریبی و با حل مرتبه صفر و مرتبه یک، فرکانس‌های بی‌بعد استخراج گردیده است. مزایای استفاده از این روش سادگی روابط و زمان اجرا مناسب در کدنویسی می‌باشد. در نهایت تاثیر پارامترهایی مانند پارامتر غیرموضعی، مقیاس طول ماده، تاثیر نوع تغییر ماده و نسبت طول به ضخامت بر روی ارتعاشات آزاد نانو تیر بررسی شده است.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Free vibration of axially functionally graded nanobeam with an attached mass based on nonlocal strain gradient theory via new ADM numerical method

نویسندگان [English]

  • MohammadReza Eghbali 1
  • Seyyed Amirhosein Hosseini 2
  • Omid Rahmani 3
1 Mech. Eng., Zanjan University
2 Department of Industrial, Mechanical and Aerospace Engineering, Buein Zahra Technical University, Buein Zahra, Qazvin, Iran
3 University of Zanjan
چکیده [English]

The purpose of this paper is to study the free vibrations along a longitudinal line of a targeted graded nano beam with an attached mass based on the theory of nonlocal strain gradient theory using the asymptotic development method. nowadays due to the importance and usage of nano structures, investigation and recognition of their mechanical and mechanical properties seem necessary. Due to its small size and behavior depending on its size and the inability of classical theories to predict dependant mechanical behaviors, non-classical theories have been used. The governing equations and boundary condition relations of targeted graded nano beam with five boundary conditions simplesimple, clamped-free, clamped-clamped, clampedsimple, and clamped-attached mass have derived by using Hamiltonian principle. Then differential equation is solved analytically to obtain the frequency equations for the five boundary conditions. In the following, dimensionless frequencies by using the solving zero and the first order of numerical asymptotic development method have derived. Advantage of this method is simplicity, noncomplex mathematical relations and proper coding execution time. Eventually, the effect of nonlocal parameters, material length scale, type of material, the ratio of length to thickness, and mode number on free vibration nano beam has been investigated.

کلیدواژه‌ها [English]

  • Nano beam
  • Nonlocal strain gradient theory
  • Axially functionally graded
  • Natural Frequencies
  • Asymptotic development method
[1] V.K. Varadan, L. Chen, J. Xie, Nanomedicine: design and applications of magnetic nanomaterials, nanosensors and nanosystems, John Wiley & Sons, 2008.
[2] H. Fan, S. Qin, A piezoelectric sensor embedded in a non-piezoelectric matrix, International Journal of Engineering Science, 33(3) (1995) 379-388.
[3] A. Rasooly, K.E. Herold, K.E. Herold, Biosensors and biodetection, Springer, 2009.
[4] L. Yu, G. Bottai-Santoni, V. Giurgiutiu, Shear lag solution for tuning ultrasonic piezoelectric wafer active sensors with applications to Lamb wave array imaging, International Journal of Engineering Science, 48(10) (2010) 848-861.
[5] N.V. Lavrik, M.J. Sepaniak, P.G. Datskos, Cantilever transducers as a platform for chemical and biological sensors, Review of Scientific Instruments, 75(7) (2004) 2229-2253.
[6] A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of applied physics, 54(9) (1983) 4703-4710.
[7] M. Zarepour, S.A.H. Hosseini, A.H. Akbarzadeh, Geometrically nonlinear analysis of Timoshenko piezoelectric nanobeams with flexoelectricity effect based on Eringen's differential model, Applied Mathematical Modelling, 69 (2019) 563-582.
[8] O. Rahmani, S. Deyhim, S. Hosseini, A. Hossein, Size dependent bending analysis of micro/nano sandwich structures based on a nonlocal high order theory, STEEL AND COMPOSITE STRUCTURES, 27(3) (2018) 371-388.
[9] O. Rahmani, M. Shokrnia, H. Golmohammadi, S. Hosseini, Dynamic response of a single-walled carbon nanotube under a moving harmonic load by considering modified nonlocal elasticity theory, The European Physical Journal Plus, 133(2) (2018) 42.
[10] M. Ghadiri, S. Hosseini, M. Karami, M. Namvar, In-Plane and out of Plane Free Vibration of U-Shaped AFM Probes Based on the Nonlocal Elasticity, Journal of Solid Mechanics Vol, 10(2) (2018) 285-299.
[11] M. Zarepour, S.A. Hosseini, M. Ghadiri, Free vibration investigation of nano mass sensor using differential transformation method, Appl. Phys. A, 123(3) (2017) 181.
[12] O. Rahmani, S. Norouzi, H. Golmohammadi, S. Hosseini, Dynamic response of a double, single-walled carbon nanotube under a moving nanoparticle based on modified nonlocal elasticity theory considering surface effects, Mechanics of Advanced Materials and Structures, 24(15) (2017) 1274-1291.
[13] R. Mindlin, H. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11(1) (1962) 415-448.
[14] 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, 51(8) (2003) 1477-1508.
[15] R.A. Toupin, Theories of elasticity with couple-stress, Archive for Rational Mechanics and Analysis, 17(2) (1964) 85-112.
[16] F. Yang, A. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39(10) (2002) 2731-2743.
[17] J.W. Lee, J.Y. Lee, Free vibration analysis of functionally graded Bernoulli-Euler beams using an exact transfer matrix expression, International Journal of Mechanical Sciences, 122 (2017) 1-17.
[18] F. Ebrahimi, M.R. Barati, Porosity-dependent vibration analysis of piezo-magnetically actuated heterogeneous nanobeams, Mechanical Systems and Signal Processing, 93 (2017) 445-459.
[19] X. Li, L. Li, Y. Hu, Z. Ding, W. Deng, Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Composite Structures, 165 (2017) 250-265.
[20] Z. Lv, H. Liu, Uncertainty modeling for vibration and buckling behaviors of functionally graded nanobeams in thermal environment, Composite Structures, 184 (2018) 1165-1176.
[21] H. Liu, H. Liu, J. Yang, Vibration of FG magneto-electro-viscoelastic porous nanobeams on visco-Pasternak foundation, Composites Part B: Engineering, 155 (2018) 244-256.
[22] D. Cao, Y. Gao, M. Yao, W. Zhang, Free vibration of axially functionally graded beams using the asymptotic development method, Engineering Structures, 173 (2018) 442-448.
[23] Z. Lv, Z. Qiu, J. Zhu, B. Zhu, W. Yang, Nonlinear free vibration analysis of defective FG nanobeams embedded in elastic medium, Composite Structures, 202 (2018) 675-685.
[24] A. Aria, M. Friswell, A nonlocal finite element model for buckling and vibration of functionally graded nanobeams, Composites Part B: Engineering, 166 (2019) 233-246.
[25] M. Trabelssi, S. El-Borgi, R. Fernandes, L.-L. Ke, Nonlocal free and forced vibration of a graded Timoshenko nanobeam resting on a nonlinear elastic foundation, Composites Part B: Engineering, 157 (2019) 331-349.
[26] A.I. Aria, T. Rabczuk, M.I. Friswell, A finite element model for the thermo-elastic analysis of functionally graded porous nanobeams, European Journal of Mechanics-A/Solids,  (2019).
[27] H.B. Khaniki, On vibrations of FG nanobeams, International Journal of Engineering Science, 135 (2019) 23-36.
[28] H. Liu, Z. Lv, H. Wu, Nonlinear free vibration of geometrically imperfect functionally graded sandwich nanobeams based on nonlocal strain gradient theory, Composite Structures, 214 (2019) 47-61.
[29] M. Ghadiri, A. Jafari, A Nonlocal First Order Shear Deformation Theory for Vibration Analysis of Size Dependent Functionally Graded Nano beam with Attached Tip Mass: an Exact Solution, Journal of Solid Mechanics Vol, 10(1) (2018) 23-37.
[30] T. Aksencer, M. Aydogdu, Vibration of a rotating composite beam with an attached point mass, Composite Structures, 190 (2018) 1-9.
[31] C. Lim, G. Zhang, J. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, Journal of the Mechanics and Physics of Solids, 78 (2015) 298-313.
[32] S.S. Rao, Mechanical Vibrations Laboratory Manual, Year, Edition Addison-Wesley Publishing Company, 1995.
[33] G. Lütjering, J.C. Williams, Titanium, Springer Science & Business Media, 2007.
[34] J. Reddy, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science, 45(2) (2007) 288-307.