Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Nonlinear free vibration analysis of a composite beam reinforced by carbon nanotubes

  • M., Alimoradzadeh (Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University) ;
  • S.D., Akbas (Bursa Technical University, Department of Civil Engineering, Mimar Sinan Campus)
  • Received : 2021.05.17
  • Accepted : 2022.10.31
  • Published : 2023.02.10
⟨ Previous Next ⟩

Abstract

This investigation presents nonlinear free vibration of a carbon nanotube reinforced composite beam based on the Von Kármán nonlinearity and the Euler-Bernoulli beam theory The material properties of the structure is considered as made of a polymeric matrix by reinforced carbon nanotubes according to different material distributions. The governing equations of the nonlinear vibration problem is delivered by using Hamilton's principle and the Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. The nonlinear natural frequency and the nonlinear free response of the system is obtained with the effect of different patterns of reinforcement.

Keywords

References

  1. Akbas, S.D. (2013), "Free vibration characteristics of edge cracked functionally graded beams by using finite element method", Int. J. Eng. Trends Technol., 4(10), 4590-4597.
  2. Akbas, S.D. (2014), "Free vibration of axially functionally graded beams in thermal environment", International Journal of Eng. Appl. Sci., 6(3), 37-51. https://doi.org/10.24107/ijeas.251224.
  3. Akbas, S.D. (2016), "Static analysis of a nano plate by using generalized differential quadrature method", Int. J. Eng. Appl. Sci., 8(2), 30-39. https://doi.org/10.24107/ijeas.252143.
  4. Akbas, S.D. (2018a), "Nonlinear thermal displacements of laminated composite beams", Couple. Syst. Mech., 7(6), 691-705. https://doi.org/10.12989/csm.2018.7.6.691.
  5. Akbas, S.D. (2018b), "Geometrically nonlinear analysis of a laminated composite beam", Struct. Eng. Mech. 66(1), 27-36. http://doi.org/10.12989/sem.2018.66.1.027.
  6. Akbas, S.D. (2018c), "Post-buckling responses of a laminated composite beam", Steel Compos. Struct., 26(6), 733-743. http://doi.org/10.12989/scs.2018.26.6.733.
  7. Akbas, S.D. (2018d), "Thermal post-buckling analysis of a laminated composite beam", Struct. Eng. Mech., 67(4), 337-346. http://dx.doi.org/10.12989/sem.2018.67.4.337.
  8. Akbas, S.D. (2019a), "Hygrothermal post-buckling analysis of laminated composite beams", Int. J. Appl. Mech., 11(01), 1950009. https://doi.org/10.1142/S1758825119500091.
  9. Akbas, S.D. (2019b), "Nonlinear static analysis of laminated composite beams under hygro-thermal effect", Struct. Eng. Mech., 72(4), 433-441. http://dx.doi.org/10.12989/sem.2019.72.4.433.
  10. Akbas, S.D. (2019c) "Longitudinal forced vibration analysis of porous a nanorod", Muhendislik Bilimleri ve Tasarim Dergisi, 7(4), 736-743. https://doi.org/10.21923/jesd.553328.
  11. Akbas, S.D. (2019d) "Axially Forced Vibration Analysis of Cracked a Nanorod", J. Comput. Appl. Mech., 5(2), 477-485. https://doi.org/10.22059/JCAMECH.2019.281285.392.
  12. Akbas, S.D. (2020a), "Modal analysis of viscoelastic nanorods under an axially harmonic load", Adv. Nano Res., 8(4), 277-282. https://doi.org/10.12989/anr.2020.8.4.277.
  13. Akbas, S.D. (2020b), "Dynamic responses of laminated beams under a moving load in thermal environment", Steel Compos. Struct., 35(6), 729-737. https://doi.org/10.12989/scs.2020.35.6.729.
  14. Akbas, S.D. (2020c), "Dynamic analysis of a laminated composite beam under harmonic load", Couple. Syst. Mech., 9(6) http://doi.org/10.12989/csm.2020.9.6.563
  15. Akbas, S.D. (2021), "Dynamic analysis of axially functionally graded porous beams under a moving load", Steel Compos. Struct., 39(6), 811-821. https://doi.org/10.12989/scs.2021.39.6.811.
  16. Akbas, S.D. (2022), "Moving-load dynamic analysis of AFG beams under thermal effect", Steel Compos. Struct., 42(5), 649-655. https://doi.org/10.12989/scs.2022.42.5.649.
  17. Al-Furjan, M.S.H., Habibi, M., Chen, G., Safarpour, H., Safarpour, M. and Tounsi, A. (2020a), "Chaotic simulation of the multi-phase reinforced thermo-elastic disk using GDQM", Eng. Comput., 1-24.
  18. Al-Furjan, M.S.H., Habibi, M., Ni, J. and Tounsi, A. (2020b), "Frequency simulation of viscoelastic multi-phase reinforced fully symmetric systems", Eng. Comput., 1-17.
  19. Al-Furjan, M.S.H., Habibi, M., Sadeghi, S., Safarpour, H., Tounsi, A. and Chen, G. (2020c), "A computational framework for propagated waves in a sandwich doubly curved nanocomposite panel", Eng. Comput.,
  20. Al-Furjan, M.S.H., Habibi, M., Ghabussi, A., Safarpour, H., Safarpour, M. and Tounsi, A. (2021a), "Non-polynomial framework for stress and strain response of the FG-GPLRC disk using three-dimensional refined higher-order theory", Eng. Struct., 228, 111496.
  21. Al-Furjan, M.S.H., Habibi, M., Shan, L. and Tounsi, A. (2021b), "On the vibrations of the imperfect sandwich higher-order disk with a lactic core using generalize differential quadrature method", Compos. Struct., 257, 113150.
  22. Alimirzaei, S., Mohammadimehr, M. and Tounsi, A. (2019), "Nonlinear analysis of viscoelastic micro-composite beam with geometrical imperfection using FEM: MSGT electro-magneto-elastic bending, buckling and vibration solutions", Struct. Eng. Mech, 71(5), 485-502. https://doi.org/10.12989/sem.2019.71.5.485
  23. Alimoradzadeh, M., Salehi, M. and Esfarjani, S.M. (2019), "Nonlinear dynamic response of an axially functionally graded (AFG) beam resting on nonlinear elastic foundation subjected to moving load", Nonlinear Eng., 8(1), 250-260. https://doi.org/10.1515/nleng-2018-0051.
  24. Alimoradzadeh, M., Salehi, M. and Esfarjani, S.M. (2020), "Nonlinear vibration analysis of axially functionally graded microbeams based on nonlinear elastic foundation using modified couple stress theory", Periodica Polytechnica Mech. Eng., 64(2), 97-108. https://doi.org/10.3311/PPme.11684.
  25. Alimoradzadeh, M., Akbas, S.D. and Esfrajani, S.M. (2021), "Nonlinear dynamic and stability of a beam resting on the nonlinear elastic foundation under thermal effect based on the finite strain theory", Struct. Eng. Mech., 80(3), 275-284. https://doi.org/10.12989/sem.2021.80.3.275.
  26. Alimoradzadeh M. and Akbas S.D. (2021a), "Superharmonic and subharmonic resonances of atomic force microscope subjected to crack failure mode based on the modified couple stress theory", Europ. Phys. J. Plus, 136, 536. https://doi.org/10.1140/epjp/s13360-021-01539-0.
  27. Alimoradzadeh, M., Akbas, S.D. and Esfrajani, S.M. (2021b), "Nonlinear dynamic and stability of a beam resting on the nonlinear elastic foundation under thermal effect based on the finite strain theory", Struct. Eng. Mech., 80(3), 275-284. https://doi.org/10.12989/sem.2021.80.3.275.
  28. Alimoradzadeh M. and Akbas S.D. (2022a), "Superharmonic and subharmonic resonances of a carbon nanotube-reinforced composite beam", Adv. Nano Res., 12(4), 353-363. https://doi.org/10.12989/anr.2022.12.4.353.
  29. Alimoradzadeh M. and Akbas S.D. (2022b), "Nonlinear dynamic behavior of functionally graded beams resting on nonlinear viscoelastic foundation under moving mass in thermal environment", Struct. Eng. Mech., 81(6), 705-714. https://doi.org/10.12989/sem.2022.81.6.705.
  30. Alimoradzadeh M. and Akbas S.D. (2022c), "Nonlinear dynamic responses of cracked atomic force microscopes", Struct. Eng. Mech., 82(6), 747-756. https://doi.org/10.12989/sem.2022.82.6.747.
  31. Alimoradzadeh M. and Akbas S.D. (2022d), "Nonlinear thermal vibration of FGM beams resting on nonlinear viscoelastic foundation", Steel Compos. Struct., 44(4), 543-553. https://doi.org/ 10.12989/scs.2022.44.4.557.
  32. Ansari, M., Esmailzadeh, E. and Younesian, D. (2010), "Internal-external resonance of beams on non-linear viscoelastic foundation traversed by moving load", Nonlinear Dyn., 61(1), 163-182. Doi: https://doi.org/10.1007/s11071-009-9639-0
  33. Arshid, E., Khorasani, M., Soleimani-Javid, Z., Amir, S., & Tounsi, A. (2021). Porosity-dependent vibration analysis of FG microplates embedded by polymeric nanocomposite patches considering hygrothermal effect via an innovative plate theory. Engineering with Computers, 1-22.
  34. Babu Arumugam, A., Rajamohan, V., Bandaru, N., Sudhagar P.E. and Kumbhar, S.G. (2019), "Vibration analysis of a carbon nanotube reinforced uniform and tapered composite beams", Archives Acoustics, 309-320. Doi :http://dx.doi.org/10.24425/aoa.2019.128494.
  35. Bakoura, A., Bourada, F., Bousahla, A.A., Tounsi, A., Benrahou, K. H., Tounsi, A. and Mahmoud, S.R. (2021), "Buckling analysis of functionally graded plates using HSDT in conjunction with the stress function method", Comput. Concrete, 27(1), 73-83. https://doi.org/10.12989/CAC.2021.27.1.073
  36. Bendenia, N., Zidour, M., Bousahla, A.A., Bourada, F., Tounsi, A., Benrahou, K.H. and Tounsi, A. (2020), "Deflections, stresses and free vibration studies of FG-CNT reinforced sandwich plates resting on Pasternak elastic foundation", Comput. Concrete, 26(3), 213-226. https://doi.org/10.12989/CAC.2020.26.3.213
  37. Bouazza, M. and Zenkour, A.M. (2020), "Vibration of carbon nanotube-reinforced plates via refined n th-higher-order theory", Archive Appl. Mech., 90(8), 1755-1769. http://dx.doi.org/10.1007/s00419-020-01694-3
  38. Bourada, F., Bousahla, A.A., Tounsi, A., Bedia, E.A., Mahmoud, S. R., Benrahou, K.H. and Tounsi, A. (2020), "Stability and dynamic analyses of SW-CNT reinforced concrete beam resting on elastic-foundation", Comput. Concrete, 25(6), 485-495. https://doi.org/10.12989/CAC.2020.25.6.485
  39. Bousahla, A.A., Bourada, F., Mahmoud, S.R., Tounsi, A., Algarni, A., Bedia, E.A. and Tounsi, A. (2020), "Buckling and dynamic behavior of the simply supported CNT-RC beams using an integral-first shear deformation theory", Comput. Concrete, 25(2), 155-166. https://doi.org/10.12989/cac.2020.25.2.155
  40. Civalek, O., Uzun, B., Yayli, M.O. and Akgoz, B. (2020), "Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method", Europ. Phys. J. Plus, 135(4), 381. https://doi.org/10.1140/epjp/s13360-020-00385-w.
  41. Ebrahimi, F., Shaghaghi, G.R. and Boreiry, M. (2016), "An investigation into the influence of thermal loading and surface effects on mechanical characteristics of nanotubes", Struct Eng Mech, 57(1), 179-200. http://dx.doi.org/10.12989/sem.2016.57.1.179.
  42. Fattahi, A.M. and Safaei, B. (2017), "Buckling analysis of CNT-reinforced beams with arbitrary boundary conditions", Microsyst. Technol., 23(10), 5079-5091. http://dx.doi.org/10.1007/s00542-017-3345-5.
  43. Fernandes, R., Mousavi, S.M. and El-Borgi, S. (2016), "Free and forced vibration nonlinear analysis of a microbeam using finite strain and velocity gradients theory", Acta Mechanica, 227(9), 2657-2670. https://doi.org/10.1007/s00707-016-1646-x.
  44. Ghayesh, M.H. (2009), "Stability characteristics of an axially accelerating string supported by an elastic foundation", Mech. Machine Theory, 44(10), 1964-1979. https://doi.org/10.1016/j.mechmachtheory.2009.05.004
  45. Ghayesh, M.H. (2012), "Nonlinear dynamic response of a simply-supported Kelvin-Voigt viscoelastic beam, additionally supported by a nonlinear spring", Nonlinear Analysis: Real World Appl., 13(3), 1319-1333. https://doi.org/10.1016/j.nonrwa.2011.10.009
  46. Ghayesh, M.H., Yourdkhani, M., Balar, S. and Reid, T. (2010), "Vibrations and stability of axially traveling laminated beams", Appl. Matha. Comput., 217(2), 545-556. https://doi.org/10.1016/j.amc.2010.05.088
  47. Ghayesh, M.H., Kazemirad, S. and Darabi, M.A. (2011), "A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions", J. Sound Vib., 330(22), 5382-5400. https://doi.org/10.1016/j.jsv.2011.06.001
  48. Ghayesh, M.H., Kazemirad, S. and Reid, T. (2012), "Nonlinear vibrations and stability of parametrically exited systems with cubic nonlinearities and internal boundary conditions: A general solution procedure", Appl. Mathem. Modelling, 36(7), 3299-3311. https://doi.org/10.1016/j.apm.2011.09.084
  49. Ghayesh, M.H. (2019), "Viscoelastic nonlinear dynamic behaviour of Timoshenko FG beams", Europ. Phys. J. Plus, 134(8), 401. https://doi.org/10.1140/epjp/i2019-12472-x.
  50. Gholami, R., Ansari, R. and Gholami, Y. (2017), "Nonlinear resonant dynamics of geometrically imperfect higher-order shear deformable functionally graded carbon-nanotube reinforced composite beams", Compos. Struct., 174, 45-58. http://dx.doi.org/10.1140/epjp/i2018-12103-2.
  51. Hachemi, H., Bousahla, A.A., Kaci, A., Bourada, F., Tounsi, A., Benrahou, K.H. and Mahmoud, S.R. (2021), "Bending analysis of functionally graded plates using a new refined quasi-3D shear deformation theory and the concept of the neutral surface position", Steel Composite Struct., 39(1), 51-64. https://doi.org/10.12989/SCS.2021.39.1.051
  52. Heidari, M. and Arvin, H. (2019), "Nonlinear free vibration analysis of functionally graded rotating composite Timoshenko beams reinforced by carbon nanotubes", J. Vib. Control, 25(14), 2063-2078. http://dx.doi.org/10.1016/j.compstruct.2016.12.009.
  53. Heidari, F., Taheri, K., Sheybani, M., Janghorban, M. and Tounsi, A. (2021), "On the mechanics of nanocomposites reinforced by wavy/defected/aggregated nanotubes", Steel Compos. Struct., 38(5), 533-545. https://doi.org/10.12989/SCS.2021.38.5.533
  54. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature, 354(6348), 56-58. http://dx.doi.org/10.1038/354056a0.
  55. Ke, L.L., Yang, J. and Kitipornchai, S. (2013), "Dynamic stability of functionally graded carbon nanotube-reinforced composite beams", Mech. Adv. Mater. Struct., 20(1), 28-37. http://dx.doi.org/10.1080/15376494.2011.581412.
  56. Kirlangic, O. and Akbas, S.D. (2020), "Comparison study between layered and functionally graded composite beams for static deflection and stress analyses", J. Comput. Appl. Mech., 51(2), 294-301. https://doi.org/10.22059/JCAMECH.2020.296319.473.
  57. Kirlangic, O. and Akbas, S.D. (2021), "Dynamic responses of functionally graded and layered composite beams", Smart Struct. Syst., 27(1), 115-122. https://doi.org/10.12989/sss.2021.27.1.115.
  58. Kiani, Y. and Mirzaei, M. (2019), "Nonlinear stability of sandwich beams with carbon nanotube reinforced faces on elastic foundation under thermal loading", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233(5), 1701-1712. https://doi.org/10.1177/0954406218772613.
  59. Lin, F. and Xiang, Y. (2014), "Numerical analysis on nonlinear free vibration of carbon nanotube reinforced composite beams", Int. J. Struct. Stab. Dyn., 14(01), 1350056. http://dx.doi.org/10.1142/S0219455413500569.
  60. Mayandi, K. and Jeyaraj, P. (2015), "Bending, buckling and free vibration characteristics of FG-CNT-reinforced polymer composite beam under non-uniform thermal load", Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 229(1), 13-28. Doi: http://dx.doi.org/10.1177/1464420713493720
  61. Medani, M., Benahmed, A., Zidour, M., Heireche, H., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2019), "Static and dynamic behavior of (FG-CNT) reinforced porous sandwich plate using energy principle", Steel Compos. Struct., 32(5), 595-610. https://doi.org/10.12989/scs.2019.32.5.595
  62. Menasria, A., Kaci, A., Bousahla, A.A., Bourada, F., Tounsi, A., Benrahou, K.H. and Mahmoud, S.R. (2020), "A four-unknown refined plate theory for dynamic analysis of FG-sandwich plates under various boundary conditions", Steel Compos. Struct., 36(3), 355-367. https://doi.org/10.12989/scs.2020.36.3.355
  63. Nayfeh, A.H., Mook, D.T. and Holmes, P. (1980), "Nonlinear oscillations", https://doi.org/10.1002/9783527617586.
  64. Rafiee, M., He, X.Q. and Liew, K.M. (2014), "Non-linear dynamic stability of piezoelectric functionally graded carbon nanotubereinforced composite plates with initial geometric imperfection", Int. J. Non-Linear Mech., 59, 37-51. https://doi.org/10.1016/j.ijnonlinmec.2013.10.011.
  65. Samadpour, M., Asadi, H. and Wang, Q. (2016), "Nonlinear aerothermal flutter postponement of supersonic laminated composite beams with shape memory alloys", Europ. J. Mech. A/Solids, 57, 18-28. https://doi.org/10.1016/j.euromechsol.2015.11.004.
  66. Shafiei, H. and Setoodeh, A.R. (2017), "Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation", Steel Compos. Struct., 24(1), 65-77. http://dx.doi.org/10.12989/scs.2017.24.1.065.
  67. Shen, H.S. (2009), "Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments", Compos. Struct., 91(1), 9-19. https://doi.org/10.1016/j.compstruct.2009.04.026.
  68. Shi, Z., Yao, X., Pang, F. and Wang, Q. (2017), "An exact solution for the free-vibration analysis of functionally graded carbon-nanotube-reinforced composite beams with arbitrary boundary conditions", Sci. Reports, 7(1), 1-18. https://doi.org/10.1038/s41598-017-12596-w.
  69. Simsek, M. (2014), "Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He's variational method", Compos. Struct., 112, 264-272. https://doi.org/10.1016/j.compstruct.2014.02.010.
  70. Tagrara, S.H., Benachour, A., Bouiadjra, M.B. and Tounsi, A. (2015), "On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams", Steel Compos. Struct., 19(5), 1259-1277. http://dx.doi.org/10.12989/scs.2015.19.5.1259
  71. Tahir, S. I., Chikh, A., Tounsi, A., Al-Osta, M. A., Al-Dulaijan, S. U. and Al-Zahrani, M.M. (2021), "Wave propagation analysis of a ceramic-metal functionally graded sandwich plate with different porosity distributions in a hygro-thermal environment", Compos. Struct., 269, 114030.
  72. Thang, P.T., Nguyen, T.T. and Lee, J. (2017), "A new approach for nonlinear buckling analysis of imperfect functionally graded carbon nanotube-reinforced composite plates", Compos. Part B: Eng., 127, 166-174. http://dx.doi.org/10.1016/j.compositesb.2016.12.002.
  73. Van Do, V. N., Jeon, J.T. and Lee, C.H. (2020), "Dynamic analysis of carbon nanotube reinforced composite plates by using Bezier extraction based isogeometric finite element combined with higher-order shear deformation theory", Mech. Mater., 142, 103307. http://dx.doi.org/10.1016/j.mechmat.2019.103307.
  74. Wattanasakulpong, N. and Ungbhakorn, V. (2013), "Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation", Comput. Mater. Sci., 71, 201-208. http://dx.doi.org/10.1016/j.commatsci.2013.01.028.
  75. Yas, M.H. and Samadi, N. (2012), "Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation", Int. J. Pressure Vessels Piping, 98, 119-128. http://dx.doi.org/10.1016/j.ijpvp.2012.07.012.
  76. Zerrouki, R., Karas, A., Zidour, M., Bousahla, A.A., Tounsi, A., Bourada, F. and Mahmoud, S.R. (2021), "Effect of nonlinear FG-CNT distribution on mechanical properties of functionally graded nano-composite beam", Struct. Eng. Mech., 78(2), 117-124.