Analysis of Nonlinear Vibration of Piezoelectric Nanobeam Embedded in Multiple Layers Elastic Media in a Thermo-Magnetic Environment Using Iteration Perturbation Method

Document Type : Research Paper

Author

Department of Mechanical Engineering, University of Lagos, Akoka, Lagos, Nigeria----- Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria

10.22034/jsm.2021.1911067.1648

Abstract

In this work, analysis of nonlinear vibration of piezoelectric nanobeam in a thermo-magnetic environment embedded in Winkler, Pasternak, quadratic and cubic nonlinear elastic media for simply supported and clamped boundary conditions is presented. With the considerations of Von- Karman geometric nonlinearity effect and with the aids of nonlocal elasticity theory as well as Euler–Bernoulli beam model, the equation of motion for the nanobeam is derived using Hamilton’s principle. The nonlinear dynamic model is solved using Galerkin-decomposition coupled with iteration perturbation method. From the parametric studies, it is shown that the frequency of the nanobeam increases at low temperatures but decreases at high temperatures. The nonlocal parameter decreases the frequencies of the piezoelectric nanobeam. An increase in the quadratic nonlinear elastic medium stiffness causes a decrease in the first mode of the nanobeam with clamped-clamped supports and an increase in all modes of the simply supported nanobeam at both low and high temperatures. When the magnetic force, cubic nonlinear elastic medium stiffness, and amplitude increase, there is an increase in all mode frequencies of the nanobeam. An increase in the temperature change at high temperature reduces the frequency ratio but at low or room temperature, an increase in temperature change, increases the frequency ratio of the structure nanotube. The significance of this study is evident in the design and applications of nanobeams in thermal and magnetic environments.

Keywords

  1.  Iijima S., 1991, Helical micro tubes of graphitic carbon, Nature 354: 56-58.
  2.  Terrones M., Banhart F., Grobert N., Charlier J., Terrones C., Ajayan H., 2002, Molecular junctions by joining single-walled carbon nanotubes, Physical Review Letters 89: 07550.
  3.  Nagy P., Ehlich R., Biro L.P., Gjyulai J., 2000, Y-branching of single walled carbon nanotubes, Applied Physics A 70: 481-483.
  4.  Chernozatonskii L.A., 1992, Carbon nanotubes connectors and planar jungle gyms, Applied Physics A 172: 173-176.
  5.  Liew K.M., Wong C.H., He X.Q., Tan M.J., Meguid S.A., 2004, Nanomechanics of single and multiwalled carbon nanotubes, Physical Review B 69: 115429.
  6.  Pantano A., Boyce M.C., Parks D.M., 2004, Mechanics of axial compression of single and multi-wall carbon nanotubes, Journal of Engineering Materials and Technology 126: 279-284.
  7.  Pantano A., Parks D.M., Boyce M.C., 2004, Mechanics of deformation of single- and multi-wall carbon nanotubes, Journal of the Mechanics and Physics of Solids 52: 789-821.
  8.  Qian D., Wagner G.J., Liu W.K., Yu M.F., Ruoff R.S., 2002, Mechanics of carbon nanotubes, Applied Mechanics Reviews 55: 495-533.
  9.  Salvetat J.P., Bonard J.-M., Thomson N.H., Kulik A.J., Forro L., Benoit W., Zuppiroli L., 1999, Mechanical properties of carbon nanotubes, Applied Physics A 69: 255-260.
  10.  Sears A., Batra R.C., 2006, Buckling of carbon nanotubes under axial compression, Physical Review B 73: 085410.
  11.  Yoon J., Ru C.Q., Mioduchowski A., 2002, Noncoaxial resonance of an isolated multiwall carbon nanotube, Physical Review B 66: 233402.
  12.  Wang X., Cai H., 2006, Effects of initial stress on non-coaxial resonance of multi-wall carbon nanotubes, Acta Materialia 54: 2067-2074.
  13.  Wang C.M., Tan V.B.C., Zhang Y.Y., 2006, Timoshenko beam model for vibration analysis of multi-walled carbon nanotubes, Journal of Sound and Vibration 294: 1060-1072.
  14.  Zhang Y., Liu G., Han X., 2005, Transverse vibrations of double-walled carbon nanotubes under compressive axial load, Physics Letters A 340: 258-266.
  15.  Elishakoff I., Pentaras D., 2009, Fundamental natural frequencies of double-walled carbon nanotubes, Journal of Sound and Vibration 322: 652-664.
  16.  Buks E., Yurke B., 2006, Mass detection with nonlinear nanomechanical resonator, Physical Review E 74: 046619.
  17.  Postma H.W.C., Kozinsky I., Husain A., Roukes M.L., 2005, Dynamic range of nanotube- and nanowire-based electromechanical systems, Applied Physics Letters 86: 223105.
  18.  Fu Y.M., Hong J.W., Wang X.Q., 2006, Analysis of nonlinear vibration for embedded carbon nanotubes, Journal of Sound and Vibration 296: 746-756.
  19.  Dequesnes M., Tang Z., Aluru N.R., 2004, Static and dynamic analysis of carbon nanotube-based switches, Transactions of the ASME 126: 230-237.
  20.  Ouakad H.M., Younis M.I., 2010, Nonlinear dynamics of electrically actuated carbon nanotube resonators, Journal of Computational and Nonlinear Dynamics 5: 011009.
  21.  Zamanian M., Khadem S.E., Mahmoodi S.N., 2009, Analysis of non-linear vibrations of a microresonator under piezoelectric and electrostatic actuations, Journal of Mechanical Engineering Science 223: 329-344.
  22. Belhadj A., Boukhalfa A., Belalia S., 2016, Carbon nanotube structure vibration based on nonlocal elasticity, Journal of Modern Materials 3(1): 9-13.
  23.  Abdel-Rahman E.M., Nayfeh A.H., 2003, Secondary resonances of electrically actuated resonant microsensors, Journal of Micromechnics and Microengineering 13: 491-501.
  24. Hawwa M.A., Al-Qahtani H.M., 2010, Nonlinear oscillations of a double-walled carbon nanotube, Computational Materials Science 48: 140-143.
  25.  Hajnayeb A., Khadem S.E., 2012, Nonlinear vibration and stability analysis of a double-walled carbon nanotube under electrostatic actuation, Journal of Sound and Vibration 331: 2443-2456.
  26.  Xu K.Y., Guo X.N., Ru C.Q., 2006, Vibration of a double-walled carbon nanotube aroused by nonlinear intertube van der Waals forces, Journal of Applied Physics 99: 064303.
  27. Lei X.W., Natsuki T., Shi J.X., Ni Q.Q., 2012, Surface effects on the vibrational frequency of double-walled carbon nanotubes using the nonlocal Timoshenko beam model, Composites Part B 43: 64-69.
  28.  Ghorbanpour A.A., Zarei M.S., Amir S., Khoddami M.Z., 2013, Nonlinear nonlocal vibration of embedded DWCNT conveying fluid using shell model, Physica B 410: 188-196.
  29.  Yoon J., Ru C.Q., Mioduchowski A., 2002, Non-coaxial resonance of an isolated multiwall carbon nanotube, Physical Review B 66: 233402-233414.
  30. Yoon J., Ru C.Q., Mioduchowski A., 2003, Vibration of an embedded multiwalled carbon nanotube [J], Composites Science and Technology 63: 1533-1542.
  31. Ansari R., Hemmatnezhad M., 2011, Nonlinear vibrations of embedded multi-walled carbon nanotubes using a variational approach, Mathematical and Computer Modelling 53(5-6): 927-938.
  32.  Ghorbanpour Arani A., Rabbani H., Amir S., Khoddami Maraghi Z., Mohammadimehr M., Haghparast E., Analysis of nonlinear vibrations for multi-walled carbon nanotubes embedded in an elastic medium, Journal of Solid Mechanics 3(3): 258-270.
  33. Yoon J., Ru C.Q.C., Miodochowski A., 2003, Vibration of an embedded multiwalled carbon nanotubes, Composites Science and Technology 63: 1533-1542.
  34. Wang C.M., Tan V.B.C., Zhang Y.Y., 2006, Timoshenko beam model for vibration analysis of multi-walled carbon nanotubes, Journal of Sound and Vibration 294: 1060-1072.
  35.  Aydogdu M., 2008, Vibration of multi-walled carbon nanotubes by generalized shear deformation theory, International Journal of Mechanical Sciences 50: 837-844.
  36.  Sobamowo M. G., 2016, Nonlinear vibration analysis of single-walled carbon nanotube conveying fluid in slip boundary bonditions using variational iterative method, Journal of Applied and Computational Mechanics 2(4): 208-221.
  37.  Sobamowo M.G., 2017, Nonlinear analysis of flow-induced vibration in fluid-conveying structures using differential transformation method with cosine-after treatment technique, Iranian Journal of Mechanical Engineering Transactions of the ISME 18(1): 5-42.
  38.  Sobamowo M.G., 2017, Nonlinear thermal and flow-induced vibration analysis of fluid-conveying carbon nanotube resting on Winkler and Pasternak foundations, Thermal Science and Engineering Progress 4: 133-149.
  39. Sobamowo M.G., Ogunmola B.Y., Osheku C.A., 2017, Thermo-mechanical nonlinear vibration analysis of fluid-conveying structures subjected to different boundary conditions using Galerkin-Newton-Harmonic balancing method, Journal of Applied and Computational Mechanics 3(1): 60-79.
  40. Arefi A., Nahvi H., 2017, Stability analysis of an embedded single-walled carbon nanotube with small initial curvature based on nonlocal theory, Mechanics of Advanced Materials and Structures 24(11): 962-970.
  41. Cigeroglu E., Samandari H., 2014, Nonlinear free vibrations of curved double walled carbon nanotubes using differential quadrature method, Physica E 64: 95-105.
  42.  Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54(9): 4703-4710.
  43.  Eringen A.C., 1972, Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science 10(5): 425-435.
  44.  Eringen A.C., 2002, Nonlocal Continuum Field Theories, Springer, New York.
  45.  Eringen A.C., Edelen D.G.B., 1972, On nonlocal elasticity, International Journal of Engineering Science 10(3): 233-248.
  46.  Yang F., Chong A., Lam D.C.C., Tong P., 2002, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39(10): 2731-
  47.  Park S., Gao X.-L., 2008, Variational formulation of a modified couple stress theory and its application to a simple shear problem, Zeitschrift für Angewandte Mathematik und Physik 59(5): 904-
  48.  Peddieson J., Buchanan G.R., McNitt R.P., 2003, Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science 41(3-5): 305-
  49.  Lu P., Lee H., Lu C., Zhang P., 2006, Dynamic properties of flexural beams using a nonlocal elasticity model, Journal of Applied Physics 99(7): 073510.
  50. Reddy J., 2007, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science 45(2-8): 288-
  51.  Reddy J., Pang S., 2008, Nonlocal continuum theories of beams for the analysis of carbon nanotubes, Journal of Applied Physics 103(2): 023511.
  52.  Lim C.W., 2010, On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: Equilibrium, governing equation and static deflection, Applied Mathematicsand Mechanics 31(1): 37-
  53.  Lim C.W., 2010, Is a nanorod (or nanotube) with a lower Young’s modulus stiffer? Is not Young’s modulus a stiffness indicator ?, Science China Physics, Mechanics & Astronomy 53(4): 712-
  54.  Hosseini S., Rahmani O., 2016, Thermomechanical vibration of curved functionally graded nanobeam based on nonlocal elasticity, Journal of Thermal Stresses 39(10): 1252-
  55. Tylikowski A., 2012, Instability of thermally induced vibrations of carbon nanotubes via nonlocal elasticity, Journal of Thermal Stresses 35(1-3): 281-
  56. Ebrahimi F., Mahmoodi F., 2018, Vibration analysis of carbon nanotubes with multiple cracks in thermal environment, Advanced Nano Research 6(1): 57-
  57.  Zhang Y., Liu X., Liu G., 2007, Thermal effect on transverse vibrations of double-walled carbon nanotubes, Nanotechnology 18(44): 445701.
  58. Murmu T., Pradhan S., 2009, Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Computational Materials Science 46(4): 854-
  59.  Karli_ci_c D., Jovanovi_c D., Kozi_c P., Caji_c M., 2015, Thermal and magnetic effects on the vibration of a cracked nanobeam embedded in an elastic medium, Journalof Mechanics of Materials and Structures 10(1): 43-
  60.  Zarepour M., Hosseini S. A., 2016, A semi analytical method for electro-thermo-mechanical nonlinear vibration analysis of nanobeam resting on the Winkler–Pasternak foundations with general elastic boundary conditions, Smart Materials and Structures 25(8): 085005.
  61.  Ke L., Xiang Y., Yang J., Kitipornchai S., 2009, Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory, Computational Materials Science 47(2): 409-
  62.  Togun N., 2016, Nonlocal beam theory for nonlinear vibrations of a nanobeam resting on elastic foundation, Boundary Value Problems 2016(1): 57.
  63.  Ansari R., Gholami R., Darabi M., 2012, Nonlinear free vibration of embedded double-walled carbon nanotubes with layerwise boundary conditions, Acta Mechanica 223(12): 2523-
  64.  Ma’en S.S., 2017, Superharmonic resonance analysis of nonlocal nano beam subjected to axial thermal and magnetic forces and resting on a nonlinear elastic foundation, Microsystem Technologies 23(8): 3319-
  65. Fallah A., Aghdam M., 2011, Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation, European Journal of Mechanics A/Solids 30(4): 571-583.
  66.  Fallah A., Aghdam M., 2012, Thermo-mechanical buckling and nonlinear free vibration analysis of functionally graded beams on nonlinear elastic foundation, Composites Part B: Engineering 43(3): 1523-1530.
  67.  Murmu T., Pradhan S., 2010, Thermal effects on the stability of embedded carbon nanotubes, Computational Materials Science 47(3): 721-726.
  68.  Simsek M., 2014, Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory, Composites Part B: Engineering 56: 621-628.
  69.  Murmu T., Pradhan S.C., 2009, Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Computational Materials Science 46: 854-859.
  70.  Abdullah S.S., Hosseini-Hashemi S., Hussein N.A., Nazemnezhad R., 2020, Thermal stress and magnetic effects on nonlinear vibration of nanobeams embedded in nonlinear elastic medium, Journal of Thermal Stresses 43(10): 1316-1332.