Nonlinear Free Vibration of Buckled Size-Dependent Functionally Graded Nanobeams Using Homotopy Perturbation Method

Ziaee, S.; Mohammadi, S. A.

doi:10.22060/ajme.2018.13562.5669

دانشگاه صنعتی امیرکبیر

تعداد نشریات	7
تعداد شماره‌ها	408
تعداد مقالات	5,454
تعداد مشاهده مقاله	5,817,622
تعداد دریافت فایل اصل مقاله	5,197,821

	Nonlinear Free Vibration of Buckled Size-Dependent Functionally Graded Nanobeams Using Homotopy Perturbation Method
AUT Journal of Mechanical Engineering
مقاله 6، دوره 2، شماره 2، اسفند 2018، صفحه 191-206 اصل مقاله (2.45 M)
نوع مقاله: Research Article
شناسه دیجیتال (DOI): 10.22060/ajme.2018.13562.5669
نویسندگان
S. Ziaee^* ¹؛ S. A. Mohammadi²
¹Department of Mechanical Engineering, College of Engineering, Yasouj University, Yasouj, Iran
²Department of Mathematics, College of Sciences, Yasouj University, Yasouj, Iran
چکیده
The present study aims at investigating nonlinear free vibration of thermally buckled functionally graded nanobeam. The nonlocal nonlinear Euler-Bernoulli beam theory as well as linear eigenmodes of a functionally graded nanobeam vibrating around the first buckling configuration are employed to derive a system of ordinary differential equations via the Galerkin method. Semi-analytical solutions are obtained based on both the homotopy perturbation method and the variational iteration method. Results show that the difference between nonlinear and linear frequencies increases with a rise in the maximum lateral initial deflection, small scale parameter value, and index of the power law. Investigating the effect of the ratio of length to thickness on the variance between the nonlinear and linear frequencies shows that the aspect ratio makes no difference on the classical ratio of nonlinear to linear frequencies although the difference between the nonlocal nonlinear and linear frequencies decreases with a rise in the aspect ratio. In contrast to the ratio of the first nonlinear frequency to the first linear one which will decrease if compressive axial load increases, the values of the compressive axial load which are beyond the load bearing capacity of the functionally graded nanobeam do not affect the ratio of the second nonlinear to linear frequencies.
کلیدواژه‌ها
Nonlinear vibration؛ Buckled functionally graded nanobeam؛ Homotopy perturbation method؛ Variational iteration method

مراجع
[1] A.S. Kanani, H. Niknam, A.R. Ohadi, M.M. Aghdam, Effect of nonlinear elastic foundation on large amplitude free and forced vibration of functionally graded beam. Composite Structures, 115(2014) 115:60-68. [2] M. Simsek, H.H. Yurtcu, Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Composite Structures, 97 (2013) 378-386. [3] J. Lei, Y. He, B. Zhang, Z. Gan, P. Zeng, Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory. International Journal of Engineering Science, 72 (2013) 36-52. [4] H. Askes, E.C. Aifantis, Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results. International Journal of Solids and Structures, 48 (2011),1962-1990. [5] R. Ansari, R. Gholami, S. Sahmani, Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Composite Structures, 94 (2011) 221-228. [6] R. Ansari, R. Gholami, M. Faghih Shojaei, V. Mohammadi, S. Sahmani, Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory. Composite Structures, 100 (2013) 385-397 [7] A.R. Setoodeh, S. Afrahim, Nonlinear dynamic analysis of FG micro-pipes conveying fluid based on strain gradient theory. Composite Structures. 116 (2014) 128–135. [8] A. Ghorbani Shenas, S. Ziaee, P. Malekzadeh, Nonlinear vibration analysis of pre-twisted functionally graded microbeams in thermal environment. Thin-Walled Structures,118 (2017) 87-104. [9] A. Ghorbanpour Arani, M. Abdollahian, R. Kolahchi, Nonlinear vibration of a nanobeam elastically bonded with a piezoelectric nanobeam via strain gradient theory. International Journal of Mechanical Sciences, 100 (2015) 32-40. [10] J.N. Reddy, Microstructure-dependent couple stress theories of functionally graded beams. Journal of the Mechanics and Physics of Solids, 59 (2011) 2382-2399. [11] A. Arbind, J.N. Reddy, Nonlinear analysis of functionally graded microstructure-dependent beams. Composite Structures, 98 (2013) 272-281. [12] M.A. Eltaher, A. Khairy, A.M. Sadoun, F.A. Omar, Static and buckling analysis of functionally graded Timoshenko nanobeams. Applied Mathematics and Computation, 229 (2014) 283–295. [13] M.A. Eltaher, S.A. Emam, F.F. Mahmoud, Static and stability analysis of nonlocal functionally graded nanobeams. Composite Structures, 96 (2013) 82–88. [14] M.A. Eltaher, S.A. Emam, F.F.Mahmoud, Free vibration analysis functionally graded size-dependent nanobeams. Applied Mathematics and Computation, 218 (2012)7406-7420. [15] M.A. Eltaher, A.E. Alshorbagy, F.F. Mahmoud, Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams. Composite Structures, 99 (2013) 193-201. [16] M.A. Eltaher, A.A. Abdelrahman, A. Al-Nabawy, M. Khater, A. Mansour, Vibration of nonlinear graduation of nano-Timoshenko beam considering the neutral axis position. Applied Mathematics and Computation, 235 (2014) 512-529. [17] B. Uymaz, Forced vibration analysis of functionally graded beams using nonlocal elasticity. Composite Structures, 105 (2013) 227-239 [18] R. Nazemnezhad, Sh. Hosseini-Hashemi, Nonlocal nonlinear free vibration of functionally graded nanobeams. Composite Structures, 110 (2014) 192-199. [19] O. Rahmani, O. Pedram, Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. International Journal of Engineering Science, 77 ( 2014) 55–70. [20] H. Niknam, M.M. Aghdam, A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation. Composite Structures, 119 (2015)452-462. [21] K. Kiani, Longitudinal and transverse instability of moving nanoscale beam-like structures made of functionally graded materials. Composite Structures, 107 (2014) 610-619. [22] S. Ziaee, Small scale effect on linear vibration of buckled size-dependent FG nanobeam. Ain Shams Engineering Journal, 6 (2015) 587-598. [23] F. Ebrahimi, E. Salari, Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method. Composites Part B: Engineering, 79 (2015) 156-169. [24] F. Ebrahimi, E. Salari, Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments. Composite Structures, 128 (2015) 363-380. [25] F. Ebrahimi, E. Salari, Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions. Composites Part B: Engineering, 78 (2015) 272-290. [26] F. Ebrahimi, E. Salari, Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment. Acta Astronautica, 113 (2015) 29-50. [27] A. Ghorbanpour Arani, V. Atabakhshian, A. Loghman, A.R. Shajari, S. Amir, Nonlinear vibration of embedded SWBNNTs based on nonlocal Timoshenko beam theory using DQ method. Physica B, 407 (2012) 2549–2555. [28] M. Trabelssi, S. El-Borgi, L.L. Ke, J.N. Reddy, Nonlocal free vibration of graded nanobeams resting on a nonlinear elastic foundation using DQM and LaDQM. Composite Structures, 176 (2017) 736-747. [29] Z. Lv, H. Liu, Uncertainty modeling for vibration and buckling behaviors of functionally graded nanobeams in thermal environment. Composite Structures, 184 (2018) 1165-1176. [30] Sh. Hosseini-Hashemi, R. Nazemnezhad, M. Bedroud, Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity. Applied Mathematical Modelling, 38 (2014)3538-3553. [31] Y.W. Zhang, J. Chen, W. Zeng, Y.Y. Teng, B. Fang, J. Zang, Surface and thermal effects of the flexural wave propagation of piezoelectric functionally graded nanobeam using nonlocal elasticity. Computational Materials Science, 97 (2015)222-226. [32] M.R. Barati, On nonlinear vibrations of flexoelectric nanobeams. International Journal of Engineering Science, 112(2017) 143–153. [33] C.W. Lim, G. Zhang, J.N. 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. [34] M. Şimşek, Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. International of Engineering Science, 105 (2016) 12-27. [35] L. Li, Y. Hu, Nonlinear bending and free vibration analyses of nonlocal strain gradient beamsmade of functionally graded material. International of Engineering Science, 107 (2016) 77-97. [36] H. Mohammadi, M. Mahzoon, Thermal effects on postbuckling of nonlinear microbeams based on the modified strain gradient theory. Composite Structures, 106 (2013) 764-776. [37] X.P. Sun, Y.Z. Hong, H.L. Dai, L. Wang, Nonlinear frequency analysis of buckled nanobeams in the presence of longitudinal magnetic field. Acta Mechanica Solida Sinica, 30 (2017) 465-473. [38] S. Sahmani, M.M. Aghdam, Nonlocal strain gradient beam model for nonlinear vibration of prebuckled and postbuckled multilayer functionally graded GPLRC nanobeams. Composite Structures, 179 (2017) 77-88. [39] V. Marinca, N. Herisanu, Nonlinear dynamical systems in engineering: Some approximate approaches. Springer, 2011. [40] H.M. Sedighi, F. Daneshmand, Nonlinear transversly vibrating beams by the Homotopy perturbation method with an auxiliary term. Journal of Applied and Computational Mechanics. 1 (2015) 1-9. [41] A. Fallah, M.M. Aghdam, Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation. European Journal of Mechanics-A/Solids, 30 (2011) 571-583. [42] T. Pribodaghi, M.T. Ahmadian, M. Fesanghary, On the Homotopy analysis method for nonlinear vibration of beams. Mechanics Research Communications, 36 (2009) 143-148. [43] M.I. Qaisi, Application of the harmonic balance principle to the nonlinear free vibration of beam. Applied Acoustics, 40 (1993) 141-151. [44] L. Azrar, R. Benamar, R.G. White, A semi-analytical approach to the nonlinear dynamic response problem of S-S and C-C beams at large vibration amplitudes, Part I: general theory and application to the single mode approach to free and forced vibration analysis. Journal of Sound and Vibration, 224 (1999)183-207. [45] M. Aydogdu, A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration. Physica E, 41 (2009) 1651–1655. [46] S.A. Emam. A general nonlocal nonlinear model for buckling of nanobeams. Applied Mathematical Modelling. 37(2013) 6929-6939. [47] C. Shu, Differential quadrature and its application in engineering. Springer-Verlag London Limited, 2000. [48] S.C. Paradhan, T. Murmu, Thermo-mechanical vibration of FGM sandwich beam under variable elastic foundations using differential quadrature method. Journal of Sound and Vibration. 321 (2009) 342-362.
آمار تعداد مشاهده مقاله: 872 تعداد دریافت فایل اصل مقاله: 604

سامانه مدیریت نشریات علمی. طراحی و پیاده سازی از سیناوب

پیوندهای مفید

آمار

Nonlinear Free Vibration of Buckled Size-Dependent Functionally Graded Nanobeams Using Homotopy Perturbation Method