Structural Engineering and Mechanics

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Non-linear transverse vibrations of tensioned nanobeams using nonlocal beam theory

  • Bagdatli, Suleyman M. (Faculty of Engineering, Department of Mechanical Engineering, Celal Bayar University)
  • Received : 2015.02.07
  • Accepted : 2015.05.28
  • Published : 2015.07.25
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Abstract

In this study, nonlinear transverse vibrations of tensioned Euler-Bernoulli nanobeams are studied. The nonlinear equations of motion including stretching of the neutral axis and axial tension are derived using nonlocal beam theory. Forcing and damping effects are included in the equations. Equation of motion is made dimensionless via dimensionless parameters. A perturbation technique, the multiple scale methods is employed for solving the nonlinear problem. Approximate solutions are applied for the equations of motion. Natural frequencies of the nanobeams for the linear problem are found from the first equation of the perturbation series. From nonlinear term of the perturbation series appear as corrections to the linear problem. The effects of the various axial tension parameters and different nonlocal parameters as well as effects of different boundary conditions on the vibrations are determined. Nonlinear frequencies are estimated; amplitude-phase modulation figures are presented for simple-simple and clamped-clamped cases.

Keywords

References

  1. Adams, G. (1995), "Critical speeds and the response of a tensioned beam on an elastic foundation to repetitive moving loads", Int. J. Mech. Sci., 37(7), 773-781. https://doi.org/10.1016/0020-7403(95)00008-L
  2. Adhikari, S., Murmu, T. and McCarthy, M.A. (2013), "Dynamic finite element analysis of axially vibrating nonlocal rods", Finite Elem. Anal. Des., 63, 42-50. https://doi.org/10.1016/j.finel.2012.08.001
  3. Aydogdu, M. (2009), "A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration", Physica E: Low-Dim. Syst. Nanostruct., 41(9), 1651-1655. https://doi.org/10.1016/j.physe.2009.05.014
  4. Bagdatli, S.M. (2015), "Non-linear vibration of nanobeams with various boundary condition based on nonlocal elasticity theory", Compos. Part B: Eng., 80, 43-52. https://doi.org/10.1016/j.compositesb.2015.05.030
  5. Bagdatli, S.M. and Uslu, B. (2015), "Free vibration analysis of axially moving beam under non-ideal conditions", Struct. Eng. Mech., 54(3), 597. https://doi.org/10.12989/sem.2015.54.3.597
  6. Bagdatli, S.M., Oz, H.R. and Ozkaya, E. (2011), "Non-linear transverse vibrations and 3:1 internal resonances of a tensioned beam on multiple supports", Math. Comput. Appl., 16(1), 203-215.
  7. Bagdatli, S.M., Oz, H.R. and Ozkaya, E. (2011), "Dynamics of axially accelerating beams with an intermediate support", J. Vib. Acoust., 133(3), 031013. https://doi.org/10.1115/1.4003205
  8. Eltaher, M.A., Abdelrahman, A.A., Al-Nabawy, A., Khater, M. and Mansour, A. (2014), "Vibration of nonlinear graduation of nano-Timoshenko beam considering the neutral axis position", Appl. Math. Comput., 235, 512-529. https://doi.org/10.1016/j.amc.2014.03.028
  9. Eringen, A.C. (1983), "On differential-equations of nonlocal elasticity and solutions of screw dislocation and surface-waves", J. Appl. Phys., 54(9), 4703-10. https://doi.org/10.1063/1.332803
  10. Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer-Verlag, Newyork.
  11. Guo, Y. and Guo, W. (2003). "Mechanical and electrostatic properties of carbon nanotubes under tensile loading and electric field", J. Appl. Phys., 36(7), 805-811.
  12. Kiani, K. (2013), "Longitudinal, transverse, and torsional vibrations and stabilities of axially moving singlewalled carbon nanotubes", Curr. Appl. Phys., 13(8), 1651-1660. https://doi.org/10.1016/j.cap.2013.05.008
  13. Kural S, and Ozkaya E, (2012), "Vibrations of an axially accelerating, multiple supported flexible beam", Struct. Eng. Mech., 44(4), 521-538. https://doi.org/10.12989/sem.2012.44.4.521
  14. Kural, S, and Ozkaya, E. (2015), "Size-dependent vibrations of a micro beam conveying fluid and resting on an elastic foundation", J. Vib. Control., doi: 10.1177/1077546315589666.
  15. Li, C., Lim, C.W., Yu, J. and Zeng, Q. (2011), "Transverse vibration of pre-tensioned nonlocal nanobeams with precise internal axial loads", Sci. China Tech. Sci., 54(8), 2007-2013. https://doi.org/10.1007/s11431-011-4479-9
  16. Lim, C.W., Li, C. and Yu, J. (2009a), "Free vibration of pre-tensioned nanobeams based on nonlocal stress theory", J. Zhejiang Univ. Sci. A, 11(1), 34-42. https://doi.org/10.1631/jzus.A0900048
  17. Lim, C.W., Li, C. and Yu, J.L. (2009b), "The effects of stiffness strengthening nonlocal stress and axial tension on free vibration of cantilever nanobeams", Int. Multis. Mech., 2(3), 223-233. https://doi.org/10.12989/imm.2009.2.3.223
  18. Lim, C.W., Li, C. and Yu, J.L. (2010), "Dynamic behaviour of axially moving nanobeams based on nonlocal elasticity approach", Acta Mechanica Sinica, 26(5), 755-765. https://doi.org/10.1007/s10409-010-0374-z
  19. Lu, P. (2007), "Dynamic analysis of axially prestressed micro nanobeam structures based on nonlocal beam theory", J. Appl. Phys., 101(7), 073504. https://doi.org/10.1063/1.2717140
  20. Lu, P., Lee, H.P., Lu, C. and Zhang, P.Q. (2007), "Application of nonlocal beam models for carbon nanotubes", Int. J. Solid. Struct., 44(16), 5289-5300. https://doi.org/10.1016/j.ijsolstr.2006.12.034
  21. Main, J.A. and Jones, N.P. (2007a), "Vibration of tensioned beams with intermediate damper. I: formulation, influence of damper location", J. Eng. Mech., 133(4), 369-378. https://doi.org/10.1061/(ASCE)0733-9399(2007)133:4(369)
  22. Main, J.A. and Jones, N.P. (2007b), "Vibration of tensioned beams with intermediate damper. I: formulation, influence of damper location", J. Eng. Mech., 133(4), 379-388. https://doi.org/10.1061/(ASCE)0733-9399(2007)133:4(379)
  23. Main, J.A. and Jones, N.P. (2007c), "Vibration of tensioned beams with intermediate damper. II: damper near a support", J. Eng. Mech., 133(4), 379-388. https://doi.org/10.1061/(ASCE)0733-9399(2007)133:4(379)
  24. Murmu, T. and Adhikari, S. (2012), "Nonlocal elasticity based vibration of initially pre-stressed coupled nanobeam systems", Eur. J. Mech. A/Solid., 34, 52-62. https://doi.org/10.1016/j.euromechsol.2011.11.010
  25. Mustapha, K.B. and Zhong, Z.W. (2010). "Free transverse vibration of an axially loaded non-prismatic single-walled carbon nanotube embedded in a two-parameter elastic medium", Comput. Mater. Sci., 50(2), 742-751. https://doi.org/10.1016/j.commatsci.2010.10.005
  26. Nayfeh, A.H. (1981), Introduction to Perturbation Techniques, John Wiley, New York, USA.
  27. Nayfeh, A.H. and Mook, D.T. (1979), Nonlinear Oscillations, John Wiley, New York, USA.
  28. Ni, Q., Li, M., Tang, M. and Wang, L. (2014), "Free vibration and stability of a cantilever beam attached to an axially moving base immersed in fluid", J. Sound Vib., 333(9), 2543-2555. https://doi.org/10.1016/j.jsv.2013.11.049
  29. Oz, H.R. (2002), "Natural Frequencies of Fluid Conveying Tensioned Pipes and Carrying a Stationary Mass under Different End Conditions", J. Sound Vib., 253(2), 507-517. https://doi.org/10.1006/jsvi.2001.4010
  30. OZ, H.R. (2003), "Natural frequencies of axially travelling tensioned beams in contact with a stationary mass", J. Sound Vib., 259(2), 445-456. https://doi.org/10.1006/jsvi.2002.5157
  31. Oz, H.R. and Boyaci, H. (2000), "Transverse vibrations of tensioned pipes conveying fluid with timedependent velocity", J. Sound Vib., 236(2), 259-276. https://doi.org/10.1006/jsvi.2000.2985
  32. Peddieson, J., Buchanan, G.R. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", Int. J. Eng. Sci., 41(3-5), 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0
  33. Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 45(2-8), 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004
  34. Sears, A. and Batra, R. (2006), "Buckling of multiwalled carbon nanotubes under axial compression", Phys. Rev. B, 73(8), 085410. https://doi.org/10.1103/PhysRevB.73.085410
  35. Sudak, L.J. (2003), "Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics", J. Appl. Phys., 94, 7281. https://doi.org/10.1063/1.1625437
  36. Thai, H.T. (2012), "A nonlocal beam theory for bending, buckling, and vibration of nanobeams", Int. J. Eng. Sci., 52, 56-64. https://doi.org/10.1016/j.ijengsci.2011.11.011
  37. Yurddas, A., Ozkaya, E. and Boyaci, H. (2013), "Nonlinear vibrations of axially moving multi-supported strings having non-ideal support conditions", Nonlin. Dyn., 73, 1223-44. https://doi.org/10.1007/s11071-012-0650-5
  38. Yurddas, A., Ozkaya, E. and Boyaci, H. (2014), "Nonlinear vibrations and stability analysis of axially moving strings having non-Ideal mid-support conditions", J. Vib. Control, 20(4), 518-534. https://doi.org/10.1177/1077546312463760
  39. Wang, Q. and Liew, K.M. (2007), "Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures", Phys. Lett. A, 363(3), 236-242. https://doi.org/10.1016/j.physleta.2006.10.093
  40. Wang, Y.Z. and Li, F.M. (2014). "Nonlinear primary resonance of nano beam with axial initial load by nonlocal continuum theory", Int. J. Nonlin. Mech., 61, 74-79. https://doi.org/10.1016/j.ijnonlinmec.2014.01.008

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