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Nonlinear dynamic responses of cracked atomic force microscopes

  • Alimoradzadeh, M. (Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University) ;
  • Akbas, S.D. (Department of Civil Engineering, Bursa Technical University)
  • Received : 2021.06.10
  • Accepted : 2022.04.03
  • Published : 2022.06.25

Abstract

This study presents the nonlinear free and forced vibrations of a cracked atomic force microscopy (AFM) cantilever by using the modified couple stress. The cracked section of the AFM cantilever is considered and modeled as rotational spring. In the frame work of Euler-Bernoulli beam theory, Von-Karman type of geometric nonlinear equation and the modified couple stress theory, the nonlinear equation of motion for the cracked AFM is derived by Hamilton's principle and then discretized by using the Galerkin's method. The semi-inverse method is utilized for analysis nonlinear free oscillation of the system. Then the method of multiple scale is employed to investigate primary resonance of the system. Some numerical examples are presented to illustrate the effects of some parameters such as depth of the crack, length scale parameter, Tip-Mass, the magnitude and the location of the external excitation force on the nonlinear free and forced vibration behavior of the system.

Keywords

References

  1. Ahmadi, M., Ansari, R. and Darvizeh, M. (2019), "Free and forced vibrations of atomic force microscope piezoelectric cantilevers considering tip-sample nonlinear interactions", Thin Wall. Struct., 145, 106382. https://doi.org/10.1016/j.tws.2019.106382.
  2. Akbas S.D. (2016a), "Analytical solutions for static bending of edge cracked micro beams", Struct. Eng. Mech., 59, 579-599. https://doi.org/10.12989/sem.2016.59.3.579.
  3. Akbas, S.D. (2013a), "Geometrically nonlinear static analysis of edge cracked Timoshenko beams composed of functionally graded material", Math. Prob. Eng., 2013, Article ID 871815. https://doi.org/10.1155/2013/871815.
  4. Akbas, S.D. (2013b), "Free vibration characteristics of edge cracked functionally graded beams by using finite element method", Int. J. Eng. Trend. Technol., 4(10), 4590-4597.
  5. Akbas, S.D. (2014), "Wave propagation analysis of edge cracked circular beams under impact force", PloS one, 9(6), e100496. https://doi.org/10.1371/journal.pone.0100496.
  6. Akbas, S.D. (2015), "Large deflection analysis of edge cracked simple supported beams", Struct. Eng. Mech., 54(3), 433-451. https://doi.org/10.12989/sem.2015.54.3.433.
  7. Akbas, S.D. (2016b), "Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium", Smart Struct. Syst., 18(6), 1125-1143. https://doi.org/10.12989/sss.2016.18.6.1125.
  8. Akbas, S.D. (2016c), "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.
  9. Akbas, S.D. (2016d), "Post-buckling analysis of edge cracked columns under axial compression loads", Int. J. Appl. Mech., 8(08), 1650086. https://doi.org/10.1142/S1758825116500861.
  10. Akbas, S.D. (2017a), "Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory", Int. J. Struct. Stab. Dyn., 17, 1750033. https://doi.org/10.1142/S021945541750033X.
  11. Akbas, S.D. (2017b), "Stability of a non-homogenous porous plate by using generalized differantial quadrature method", Int. J. Eng. Appl. Sci., 9(2), 147-155. https://doi.org/10.24107/ijeas.322375.
  12. Akbas, S.D. (2017c), "Forced vibration analysis of functionally graded nanobeams", Int. J. Appl. Mech., 9(07), 1750100. https://doi.org/10.1142/S1758825117501009.
  13. Akbas, S.D. (2018a), "Forced vibration analysis of cracked functionally graded microbeams", Adv. Nano Res., 6, 39. http://doi.org/10.12989/anr.2018.6.1.039.
  14. Akbas, S.D. (2018b), "Forced vibration analysis of cracked nanobeams", J. Brazil. Soc. Mech. Sci. Eng., 40, 1-11. https://doi.org/10.1007/s40430-018-1315-1.
  15. Akbas, S.D. (2018c), "Bending of a cracked functionally graded nanobeam", Adv. Nano Res., 6(3), 219-242. https://doi.org/10.12989/anr.2018.6.3.219.
  16. Akbas, S.D. (2018d), "Geometrically nonlinear analysis of functionally graded porous beams", Wind Struct., 27(1), 59-70. https://doi.org/10.12989/was.2018.27.1.059.
  17. Akbas, S.D. (2019a), "Axially forced vibration analysis of cracked a nanorod", J. Comput. Appl. Mech, 50, 63-68. https://doi.org/10.22059/jcamech.2019.281285.392.
  18. Akbas, S.D. (2019b), "Longitudinal forced vibration analysis of porous a nanorod", Muhendislik Bilimleri ve Tasarim Dergisi, 7(4), 736-743. https://doi.org/10.21923/jesd.553328.
  19. Akbas, S.D. (2019c), "Nonlinear behavior of fiber reinforced cracked composite beams", Steel Compos. Struct., 30(4), 327-336. https://doi.org/10.12989/scs.2019.30.4.327.
  20. Akbas, S.D. (2019d), "Post-buckling analysis of a fiber reinforced composite beam with crack", Eng. Fract. Mech., 212, 70-80. https://doi.org/10.1016/j.engfracmech.2019.03.007.
  21. Akbas, S.D. (2020), "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.
  22. Akbas, S.D., Yayli, M.O., Deliktas, B. and Uzun, B. (2022), Vibration Analysis of Cracked Microbeams by Using Finite Element Method, Handbook of Damage Mechanics, Springer, Cham.
  23. Al-Furjan, M.S.H., Habibi, M., Sadeghi, S., Safarpour, H., Tounsi, A. and Chen, G. (2020), "A computational framework for propagated waves in a sandwich doubly curved nanocomposite panel", Eng. Comput., 1-18. https://doi.org/10.1007/s00366-020-01130-8.
  24. 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", Nonlin. Eng., 28, 250-260. https://doi.org/10.1515/nleng-2018-0051.
  25. Alimoradzadeh, M. and Akbas, S.D. (2021), "Superharmonic and subharmonic resonances of atomic force microscope subjected to crack failure mode based on the modified couple stress theory", Eur. Phys. J. Plus, 136(5), 1-20. https://doi.org/10.1140/epjp/s13360-021-01539-0.
  26. Alimoradzadeh, M. and Akbas, S.D. (2022), "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.
  27. 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.
  28. 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, 97-108. https://doi.org/10.3311/PPme.11684.
  29. Alimoradzadeh, M.M., Salehi, M.M. and Esfarjani, S.M.M. (2017), "Vibration analysis of FG micro-beam based on the third order shear deformation and modified couple stress theories", J. Simul. Anal. Novel Technol. Mech. Eng., 10, 51-66.
  30. 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.
  31. Caddemi, S. and Morassi, A. (2007), "Crack detection in elastic beams by static measurements", Int. J. Solid. Struct., 44, 5301-15. https://doi.org/10.1016/j.ijsolstr.2006.12.033.
  32. Chang, W.J., Yang, Y.C. and Lee, H.L. (2013), "Dynamic behaviour of atomic force microscope-based nanomachining based on a modified couple stress theory", Micro Nano Lett., 8, 832-835. https://doi.org/10.1049/mnl.2013.0493.
  33. Chang, W.J., Yang, Y.C. and Lee, H.L. (2015), "Nanomachining analysis of a multi-cracked atomic force microscope cantilever based on a modified couple stress theory", Modern Phys. Lett. B, 29, 1550186. https://doi.org/10.1142/S0217984915501869.
  34. Chen, W., Wang, L. and Dai, H. (2019), "Nonlinear free vibration of nanobeams based on nonlocal strain gradient theory with the consideration of thickness-dependent size effect", J. Mech. Mater. Struct., 14, 119-137. https://doi.org/10.2140/jomms.2019.14.119.
  35. Chorsi, M.T., Azizi, S. and Bakhtiari-Nejad, F. (2017), "Nonlinear dynamics of a functionally graded piezoelectric micro-resonator in the vicinity of the primary resonance", J. Vib. Control, 23, 400-413. https://doi.org/10.1142/S0217984915501869.
  36. Fattahi, A.M., Sahmani, S. and Ahmed, N.A. (2019), "Nonlocal strain gradient beam model for nonlinear secondary resonance analysis of functionally graded porous micro/nano-beams under periodic hard excitations", Mech. Bas. Des. Struct. Mach., 48(4), 403-432. https://doi.org/10.1080/15397734.2019.1624176.
  37. Ghayesh, M.H. and Amabili, M. (2013b), "Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support", Mech. Mach. Theory, 67, 1-16. https://doi.org/10.1016/j.ijengsci.2013.05.006.
  38. Ghayesh, M.H., Amabili, M. and Farokhi, H. (2013a), "Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory", Int. J. Eng. Sci., 63, 52-60. https://doi.org/10.1016/j.mechmachtheory.2013.03.007.
  39. Ghayesh, M.H., Farokhi, H. and Amabili, M. (2013), "Nonlinear dynamics of a microscale beam based on the modified couple stress theory", Compos. Part B: Eng., 50, 318-324. https://doi.org/10.1016/j.ijengsci.2012.12.001.
  40. Ghayesh, M.H., Farokhi, H. and Amabili, M. (2013c), "Nonlinear behaviour of electrically actuated MEMS resonators", Int. J. Eng. Sci., 71, 137-155. https://doi.org/10.1016/j.ijnonlinmec.2012.05.003.
  41. Gutierrez Gutierrez, A., Cortes Zapata, D. and Castro Guevara, D.A. (2019), "Saddle-node bifurcation and homoclinic persistence in AFMs with periodic forcing", Math. Prob. Eng., 2019, Article ID 8925687. https://doi.org/10.1155/2019/8925687.
  42. Hassannejad, R. and Amiri Jahed, S. (2018), "Nonlinear dynamic analysis of cracked micro-beams below and at the onset of dynamic pull-in instability", J. Solid Mech., 10, 110-123.
  43. He, J.H. (2007), "Variational approach for nonlinear oscillators", Chaos Soliton Fract., 34, 1430-1439. https://doi.org/10.1016/j.chaos.2006.10.026.
  44. Huang, X., Hao, H., Oslub, K., Habibi, M. and Tounsi, A. (2021), "Dynamic stability/instability simulation of the rotary size-dependent functionally graded microsystem", Eng. Comput., 1-17. https://doi.org/10.1007/s00366-021-01399-3.
  45. Jazi, M.M., Ghayour, M., Ziaei-Rad, S. and Miandoab, E.M. (2018), "Effect of size on the dynamic behaviors of atomic force microscopes", Microsyst. Technol., 24, 1755-1765. https://doi.org/10.1007/s00542-017-3698-9.
  46. Jeong, B., Cho, H., Yu, M.F., Vakakis, A.F., McFarland, D.M. and Bergman, L.A. (2013), "Modeling and measurement of geometrically nonlinear damping in a microcantilever-nanotube system", ACS nano, 7, 8547-8553. https://doi.org/10.1177/1077546315580051.
  47. Kahrobaiyan, M.H., Rahaeifard, M. and Ahmadian, M.T. (2011), "Nonlinear dynamic analysis of a V-shaped microcantilever of an atomic force microscope", Appl. Math. Model., 35, 5903-5919. https://doi.org/10.1063/1.3573390.
  48. Karami, B., Janghorban, M. and Tounsi, A. (2020), "Novel study on functionally graded anisotropic doubly curved nanoshells", Eur. Phys. J. Plus, 135, 103. https://doi.org/10.1140/epjp/s13360-019-00079-y.
  49. Khorshidi, M.A., Shaat, M., Abdelkefi, A. and Shariati, M. (2017), "Nonlocal modeling and buckling features of cracked nanobeams with von Karman nonlinearity", Appl. Phys. A, 123, 1-12. https://doi.org/10.1007/s00339-016-0658-7.
  50. Khosravi, F., Hosseini, S.A. and Tounsi, A. (2020), "Torsional dynamic response of viscoelastic SWCNT subjected to linear and harmonic torques with general boundary conditions via Eringen's nonlocal differential model", Eur. Phys. J. Plus, 135, 183. https://doi.org/10.1140/epjp/s13360-020-00207-z.
  51. Kong, S., Zhou, S., Nie, Z. and Wang, K. (2008), "The size-dependent natural frequency of Bernoulli-Euler micro-beams", Int. J. Eng. Sci., 46, 427-437. https://doi.org/10.1016/j.ijengsci.2007.10.002.
  52. Korayem, M.H. and Ebrahimi, N. (2011), "Nonlinear dynamics of tapping-mode atomic force microscopy in liquid", J. Appl. Phys., 109, 084301. https://doi.org/10.1016/j.compositesb.2013.02.021.
  53. Korayem, M.H. and Korayem, A.H. (2017), "Modeling of AFM with a piezoelectric layer based on the modified couple stress theory with geometric discontinuities", Appl. Math. Model., 45, 439-456. https://doi.org/10.1016/j.apm.2017.01.008.
  54. Laura, P.A., Pombo, J.L. and Susemihl, E.A. (1974), "A note on the vibrations of a clamped-free beam with a mass at the free end", J. Sound Vib., 37, 161-168. https://doi.org/10.1016/S0022-460X(74)80325-1.
  55. Liang, L.N., Ke, L.L., Wang, Y.S., Yang, J. and Kitipornchai, S. (2015), "Flexural vibration of an atomic force microscope cantilever based on modified couple stress theory", Int. J. Struct. Stab. Dyn., 15, 1540025. https://doi.org/10.1142/S0219455415400258.
  56. Mohammadimehr, M., Mohammadi Hooyeh, H., Afshari, H. and Salarkia, M.R. (2016), "Size-dependent effects on the vibration behavior of a Timoshenko microbeam subjected to pre-stress loading based on DQM", Mech. Adv. Compos. Struct., 3, 99-112. https://doi.org/10.22075/MACS.2016.472.
  57. Nayfeh, A.H., Mook, D.T. and Holmes, P. (1980), "Nonlinear oscillations", ASME J. Appl. Mech., 47(3), 692. https://doi.org/10.1115/1.3153771.
  58. Nazemnezhad, R. and Hosseini-Hashemi, S. (2014), "Nonlocal nonlinear free vibration of functionally graded nanobeams", Compos. Struct., 110, 192-199. https://doi.org/10.1016/j.compstruct.2013.12.006.
  59. Norouzi, H. and Younesian, D. (2015), "Chaotic vibrations of beams on nonlinear elastic foundations subjected to reciprocating loads", Mech. Res. Commun., 69, 121-128. https://doi.org/10.1016/j.mechrescom.2015.07.001.
  60. Rahi, A. (2018), "Crack mathematical modeling to study the vibration analysis of cracked micro beams based on the MCST", Microsyst. Technol., 24, 3201-3215. https://doi.org/10.1007/s00542-018-3768-7.
  61. Rahi, A. and Petoft, H. (2018), "Free vibration analysis of multi-cracked micro beams based on Modified Couple Stress Theory", J. Theor. Appl. Vib. Acoust., 4, 205-222. https://doi.org/10.22064/TAVA.2019.89997.1113.
  62. Ramezani, S. (2012), "A micro scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory", Int. J. Nonlin. Mech., 47, 863-873. https://doi.org/10.1007/s00707-012-0622-3.
  63. Rao, S.S. (2007), Vibration of Continuous Systems, Wiley, New York
  64. Rastehkenari, S.F. and Ghadiri, M. (2019), "Size-dependent random vibration analysis of AFM probe with tip mass considering surface viscoelastic effect", Eur. Phys. J. Plus, 134, 1-12. https://doi.org/10.1140/epjp/i2019-12924-3.
  65. Rega, G. and Settimi, V. (2013), "Bifurcation, response scenarios and dynamic integrity in a single-mode model of noncontact atomic force microscopy", Nonlin. Dyn., 73, 101-123. https://doi.org/10.1021/nn402479d.
  66. Rezazadeh, G., Vahdat, A.S., Tayefeh-rezaei, S. and Cetinkaya, C. (2012), "Thermoelastic damping in a micro-beam resonator using modified couple stress theory", Acta Mechanica, 223, 1137-1152. https://doi.org/10.1007/S11071-013-0771-5.
  67. Rouabhia, A., Heireche, H., Khelifi, S., Sahouane, N., Dabou, R., Ziane, A. and Tounsi, A. (2020), "Physical stability response of a SLGS resting on viscoelastic medium using nonlocal integral first-order theory", ICREATA'21, 180.
  68. Rupp, D., Rabe, U., Hirsekorn, S. and Arnold, W. (2007), "Nonlinear contact resonance spectroscopy in atomic force microscopy", J. Phys. D: Appl. Phys., 40, 7136. https://doi.org/10.1016/j.apm.201105.039.
  69. Rutzel, S., Lee, S.I. and Raman, A. (2003), "Nonlinear dynamics of atomic-force-microscope probes driven in Lennard-Jones potentials", Pro. Roy. Soc. London Ser. A: Math. Phys. Eng. Sci., 459, 1925-1948. https://doi.org/10.1063/1.1458056.
  70. Shahani, A.R., Rezazadeh, G. and Rahmani, A. (2018), "Crack influences on the static and dynamic characteristic of a micro-beam subjected to electro statically loading", J. Solid Mech., 10, 603-620.
  71. Sourki, R. and Hosseini, S.A. (2017), "Coupling effects of nonlocal and modified couple stress theories incorporating surface energy on analytical transverse vibration of a weakened nanobeam", Eur. Phys. J. Plus, 132, 1-14. https://doi.org/10.1140/epjp/i2017-11458-0.
  72. Togun, N. (2016), "Nonlinear vibration of nanobeam with attached mass at the free end via nonlocal elasticity theory", Microsyst. Technol., 22, 2349-2359. https://doi.org/10.1007/s00542-016-3062-5.
  73. Wang, C.C., Huang, C., Yau, H.T. and Hsu, S.X. (2013), "Bifurcation and chaos analysis of atomic force microscope system", Microsyst. Technol., 19, 1795-1805. https://doi.org/10.1007/s00542-013-1804-1.
  74. Wolf, K. and Gottlieb, O. (2002), "Nonlinear dynamics of a noncontacting atomic force microscope cantilever actuated by a piezoelectric layer", J. Appl. Phys., 91, 4701-4709. https://doi.org/10.1063/1.1458056.
  75. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based on strain gradient theory for elasticity", Int. J. Solid. Struct., 39, 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X.
  76. Yayli, M.O. (2020), "Axial vibration analysis of a Rayleigh nanorod with deformable boundaries", Microsyst. Technol., 26(8), 2661-2671. https://doi.org/10.1007/s00542-020-04808-7.
  77. Zerrouki, R., Karas, A., Zidour, M., Bousahla, A.A., Tounsi, A., Bourada, F., ... & 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. http://doi.org/10.12989/sem.2021.78.2.117.
  78. Zhang, Y. and Zhao, Y. (2007), "Nonlinear dynamics of atomic force microscopy with intermittent contact", Chaos Soliton. Fract., 34, 1021-1024. https://doi.org/10.1016/j.chaos.2006.03.125.