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Combined resonance of pulsatile flow-transporting FG nanotubes under forced excitation with movable boundary

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Abstract

The nanoscale fluid–solid dynamic interaction with movable boundary is common in the adhesion of nanoelectromechanical systems, which presents complex size-dependent inertia and stiffness nonlinearities affecting the resonance response, frequency band and bifurcation topology greatly. The size-dependent nonlinear primary resonance of fluid-conveying functionally graded nanotubes with movable boundary under the internal pulsatile nanoflow and external forced excitation is studied. The size dependency of the nanosolid behavior is modeled by nonlocal strain gradient theory coupled with surface elasticity, while that of nanofluid is characterized by slip flow model. A proposed comprehensive geometrically nonlinear model encompasses two parts: First, Zhang-Fu’s refined beam model is modified to reflect nonlinear curvature effect. Subsequently, the inertial nonlinearity caused by the movable boundary is introduced based on the assumption of “non-extensible beam.” A two-step perturbation-incremental harmonic balance method is developed to obtain the amplitude–frequency bifurcation curves. Results provide the bifurcation sets under three stiffness conditions. Parametric analysis reveals the combined resonance mechanism and discusses the influences of nonlocal stress, strain gradient effect, surface effect, slip flow effect and material gradient index on the results. It is found that the size-dependent nonlinear inertia not only plays the role of softening nonlinear behaviors but also changes the bifurcation topology.

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The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Lloyd, D.J.: Particle reinforced aluminium and magnesium matrix composites. Int. Mater. Rev. 39(1), 1–23 (1994)

    Google Scholar 

  2. Liu, H., Li, B., Liu, Y.: The inconsistency of nonlocal effect on carbon nanotube conveying fluid and a proposed solution based on local/nonlocal model. Eur. J. Mech. A-Solids 78, 103837 (2019)

    MathSciNet  MATH  Google Scholar 

  3. Naderi, A., Fakher, M., Hosseini-Hashemi, S.: On the local/nonlocal piezoelectric nanobeams: vibration, buckling, and energy harvesting. Mech. Syst. Signal Process. 151, 107432 (2021)

    Google Scholar 

  4. Lyu, Z., Yang, Y., Liu, H.: High-accuracy hull iteration method for uncertainty propagation in fluid-conveying carbon nanotube system under multi-physical fields. Appl. Math. Model. 79, 362–380 (2020)

    MathSciNet  MATH  Google Scholar 

  5. Caporale, A., Darban, H., Luciano, R.: Nonlocal strain and stress gradient elasticity of Timoshenko nano-beams with loading discontinuities. Int. J. Eng. Sci. 173, 103620 (2022)

    MathSciNet  MATH  Google Scholar 

  6. Behdad, S., Fakher, M., Hosseini-Hashemi, S.: Dynamic stability and vibration of two-phase local/nonlocal VFGP nanobeams incorporating surface effects and different boundary conditions. Mech. Mater. 153, 103633 (2021)

    Google Scholar 

  7. Ghane, M., Saidi, A.R., Bahaadini, R.: Vibration of fluid-conveying nanotubes subjected to magnetic field based on the thin-walled Timoshenko beam theory. Appl. Math. Model. 80, 65–83 (2020)

    MathSciNet  MATH  Google Scholar 

  8. Sadeghi-Goughari, M., Jeon, S., Kwon, H.J.: Fluid structure interaction of cantilever micro and nanotubes conveying magnetic fluid with small size effects under a transverse magnetic field. J. Fluid Struct. 94, 102951 (2020)

    Google Scholar 

  9. Darban, H., Fabbrocino, F., Luciano, R.: Size-dependent linear elastic fracture of nanobeams. Int. J. Eng. Sci. 157, 103381 (2020)

    MathSciNet  MATH  Google Scholar 

  10. Qing, H., Wei, L.: Linear and nonlinear free vibration analysis of functionally graded porous nanobeam using stress-driven nonlocal integral model. Commun. Nonlinear Sci. 109, 106300 (2022)

    MathSciNet  Google Scholar 

  11. Ghaffari, S.S., Ceballes, S., Abdelkefi, A.: Nonlinear dynamical responses of forced carbon nanotube-based mass sensors under the influence of thermal loadings. Nonlinear Dyn. 100(2), 1013–1035 (2020)

    Google Scholar 

  12. Zheng, S., Chen, D., Wang, H.: Size dependent nonlinear free vibration of axially functionally graded tapered microbeams using finite element method. Thin Wall Struct. 139, 46–52 (2019)

    Google Scholar 

  13. Ansari, R., Gholami, R.: Size-dependent nonlinear vibrations of first-order shear deformable magneto-electro-thermo elastic nanoplates based on the nonlocal elasticity theory. Int. J. Appl. Mech. 8(04), 1650053 (2016)

    Google Scholar 

  14. Malikan, M., Eremeyev, V.A.: On the geometrically nonlinear vibration of a piezo-flexomagnetic nanotube. Math. Method Appl. Sci. (2020). https://doi.org/10.1002/mma.6758

    Article  MATH  Google Scholar 

  15. Ebrahimi, F., Hosseini, S.H.S.: Nonlinear vibration and dynamic instability analysis nanobeams under thermo-magneto-mechanical loads: a parametric excitation study. Eng. Comput. 37(1), 395–408 (2021)

    Google Scholar 

  16. Ghadiri, M., Hosseini, S.H.S.: Parametric excitation of Euler–Bernoulli nanobeams under thermo-magneto-mechanical loads: nonlinear vibration and dynamic instability. Compos. B Eng. 173, 106928 (2019)

    Google Scholar 

  17. Ghayesh, M.H., Farajpour, A.: Nonlinear mechanics of nanoscale tubes via nonlocal strain gradient theory. Int. J. Eng. Sci. 129, 84–95 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Ansari, R., Gholami, R., Rouhi, H.: Size-dependent nonlinear forced vibration analysis of magneto-electro-thermo-elastic Timoshenko nanobeams based upon the nonlocal elasticity theory. Compos. Struct. 126, 216–226 (2015)

    Google Scholar 

  19. Rajasekaran, S., Khaniki, H.B.: Size-dependent forced vibration of non-uniform bi-directional functionally graded beams embedded in variable elastic environment carrying a moving harmonic mass. Appl. Math. Model. 72, 129–154 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Wang, J., Zhu, Y., Zhang, B., Shen, H., Liu, J.: Nonlocal and strain gradient effects on nonlinear forced vibration of axially moving nanobeams under internal resonance conditions. Appl. Math. Mech. 41(2), 261–278 (2020)

    MathSciNet  MATH  Google Scholar 

  21. Pourkiaee, S.M., Khadem, S.E., Shahgholi, M., Bab, S.: Nonlinear modal interactions and bifurcations of a piezoelectric nanoresonator with three-to-one internal resonances incorporating surface effects and van der Waals dissipation forces. Nonlinear Dyn. 88(3), 1785–1816 (2017)

    Google Scholar 

  22. Krysko, V.A., Jr., Awrejcewicz, J., Dobriyan, V., Papkova, I.V., Krysko, V.A.: Size-dependent parameter cancels chaotic vibrations of flexible shallow nano-shells. J. Sound Vib. 446, 374–386 (2019)

    Google Scholar 

  23. Dehrouyeh-Semnani, A.M., Nikkhah-Bahrami, M., Yazdi, M.R.H.: On nonlinear stability of fluid-conveying imperfect micropipes. Int. J. Eng. Sci 120, 254–271 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Esfahanian, V., Dehdashti, E., Dehrouyeh-Semnani, A.M.: Fluid-structure interaction in microchannel using Lattice Boltzmann method and size-dependent beam element. Adv. Appl. Math. Mech. 6(3), 345–358 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Dehrouyeh-Semnani, A.M., Nikkhah-Bahrami, M., Yazdi, M.R.H.: On nonlinear vibrations of micropipes conveying fluid. Int. J. Eng. Sci. 117, 20–33 (2017)

    MathSciNet  MATH  Google Scholar 

  26. Jin, Q., Ren, Y.: Dynamic instability mechanism of post-buckled FG nanotubes transporting pulsatile flow: size-dependence and local/global dynamics. Appl. Math. Model. 111, 139–159 (2022)

    MathSciNet  MATH  Google Scholar 

  27. Hosseini, S.H.S., Ghadiri, M.: Nonlinear dynamics of fluid conveying double-walled nanotubes incorporating surface effect: a bifurcation analysis. Appl. Math. Model. 92, 594–611 (2021)

    MathSciNet  MATH  Google Scholar 

  28. Mahmoudpour, E., Esmaeili, M.: Nonlinear free and forced vibration of carbon nanotubes conveying magnetic nanoflow and subjected to a longitudinal magnetic field using stress-driven nonlocal integral model. Thin Wall Struct. 166, 108134 (2021)

    Google Scholar 

  29. Ghayesh, M.H., Farajpour, A., Farokhi, H.: Effect of flow pulsations on chaos in nanotubes using nonlocal strain gradient theory. Commun Nonlinear Sci. 83, 105090 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Farajpour, A., Farokhi, H., Ghayesh, M.H.: Chaotic motion analysis of fluid-conveying viscoelastic nanotubes. Eur. J. Mech. A-Solids 74, 281–296 (2019)

    MathSciNet  MATH  Google Scholar 

  31. Pirmoradian, M., Torkan, E., Abdali, N., Hashemian, M., Toghraie, D.: Thermo-mechanical stability of single-layered graphene sheets embedded in an elastic medium under action of a moving nanoparticle. Mech. Mater. 141, 103248 (2020)

    Google Scholar 

  32. Hashemian, M., Falsafioon, M., Pirmoradian, M., Toghraie, D.: Nonlocal dynamic stability analysis of a Timoshenko nanobeam subjected to a sequence of moving nanoparticles considering surface effects. Mech. Mater. 148, 103452 (2020)

    Google Scholar 

  33. Lotfan, S., Fathi, R., Ettefagh, M.M.: Size-dependent nonlinear vibration analysis of carbon nanotubes conveying multiphase flow. Int. J. Mech. Sci. 115, 723–735 (2016)

    Google Scholar 

  34. Dehrouyeh-Semnani, A.M., Dehdashti, E., Yazdi, M.R.H., Nikkhah-Bahrami, M.: Nonlinear thermo-resonant behavior of fluid-conveying FG pipes. Int. J. Eng. Sci. 144, 103141 (2019)

    MathSciNet  MATH  Google Scholar 

  35. Ghayesh, M.H., Farajpour, A.: A review on the mechanics of functionally graded nanoscale and microscale structures. Int. J. Eng. Sci. 137, 8–36 (2019)

    MathSciNet  MATH  Google Scholar 

  36. Witvrouw, A., Mehta, A.: The use of functionally graded poly-SiGe layers for MEMS applications. In: Materials Science Forum, vol. 492–493, pp. 255–260 (2005). https://doi.org/10.4028/www.scientific.net/msf.492-493.255

  37. Tong, G., Liu, Y., Cheng, Q., Dai, J.: Stability analysis of cantilever functionally graded material nanotube under thermo-magnetic coupling effect. Eur. J. Mech. A-Solids 80, 103929 (2020)

    MathSciNet  MATH  Google Scholar 

  38. Wang, Y.Q., Wan, Y.H., Zu, J.W.: Nonlinear dynamic characteristics of functionally graded sandwich thin nanoshells conveying fluid incorporating surface stress influence. Thin Wall Struct. 135, 537–547 (2019)

    Google Scholar 

  39. Jin, Q., Ren, Y.: Nonlinear size-dependent bending and forced vibration of internal flow-inducing pre-and post-buckled FG nanotubes. Commun. Nonlinear Sci. 104, 106044 (2022)

    MathSciNet  MATH  Google Scholar 

  40. Ramini, A., Alcheikh, N., Ilyas, S., Younis, M.I.: Efficient primary and parametric resonance excitation of bistable resonators. AIP Adv. 6(9), 095307 (2016)

    Google Scholar 

  41. Dai, H.L., Abdelkefi, A., Wang, L.: Piezoelectric energy harvesting from concurrent vortex-induced vibrations and base excitations. Nonlinear Dyn. 77(3), 967–981 (2014)

    Google Scholar 

  42. Demšić, M., Uroš, M., Lazarević, A.J., Lazarević, D.: Resonance regions due to interaction of forced and parametric vibration of a parabolic cable. J. Sound Vib. 447, 78–104 (2019)

    Google Scholar 

  43. Yang, H., Xiao, F.: Instability analyses of a top-tensioned riser under combined vortex and multi-frequency parametric excitations. Ocean Eng. 81, 12–28 (2014)

    Google Scholar 

  44. Wang, D., Bai, C., Zhang, H.: Nonlinear vibrations of fluid-conveying FG cylindrical shells with piezoelectric actuator layer and subjected to external and piezoelectric parametric excitations. Compos. Struct. 248, 112437 (2020)

    Google Scholar 

  45. Mao, X.Y., Ding, H., Chen, L.Q.: Dynamics of a super-critically axially moving beam with parametric and forced resonance. Nonlinear Dyn. 89(2), 1475–1487 (2017)

    Google Scholar 

  46. Dai, H.L., Wang, L., Qian, Q., Ni, Q.: Vortex-induced vibrations of pipes conveying pulsating fluid. Ocean Eng. 77, 12–22 (2014)

    Google Scholar 

  47. Tang, Y.Q., Zhang, D.B., Gao, J.M.: Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions. Nonlinear Dyn. 83(1), 401–418 (2016)

    MathSciNet  MATH  Google Scholar 

  48. Sahoo, B., Panda, L.N., Pohit, G.: Combination, principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation. Int. J. NonLin Mech. 78, 35–44 (2016)

    Google Scholar 

  49. Mao, X.Y., Ding, H., Chen, L.Q.: Forced vibration of axially moving beam with internal resonance in the supercritical regime. Int. J. Mech. Sci. 131, 81–94 (2017)

    Google Scholar 

  50. Li, L., Zhang, Q., Wang, W., Han, J.: Nonlinear coupled vibration of electrostatically actuated clamped–clamped microbeams under higher-order modes excitation. Nonlinear Dyn. 90(3), 1593–1606 (2017)

    Google Scholar 

  51. Jin, Q., Ren, Y.: Nonlinear size-dependent dynamic instability and local bifurcation of FG nanotubes transporting oscillatory fluids. Acta Mech. Sin. 38, 521513 (2022)

    MathSciNet  Google Scholar 

  52. Mittal, K.L.: Progress in Adhesion and Adhesives. John Wiley & Sons, New York (2015)

    Google Scholar 

  53. Lorenzoni, M., Llobet, J., Perez-Murano, F.: Study of buckling behavior at the nanoscale through capillary adhesion force. Appl. Phys. Lett. 112(19), 193102 (2018)

    Google Scholar 

  54. Formica, G., Arena, A., Lacarbonara, W., Dankowicz, H.: Coupling FEM with parameter continuation for analysis of bifurcations of periodic responses in nonlinear structures. J. Comput. Nonlinear Dyn. 8(2), 021013 (2013)

    Google Scholar 

  55. Balasubramanian, P., Franchini, G., Ferrari, G., Painter, B., Karazis, K., Amabili, M.: Nonlinear vibrations of beams with bilinear hysteresis at supports: interpretation of experimental results. J. Sound Vib. 499, 115998 (2021)

    Google Scholar 

  56. Lacarbonara, W.: Nonlinear Structural Mechanics: Theory, Dynamical Phenomena and Modeling. Springer Science & Business Media, Berlin (2013)

    MATH  Google Scholar 

  57. Chen, W., Dai, H., Jia, Q., Wang, L.: Geometrically exact equation of motion for large-amplitude oscillation of cantilevered pipe conveying fluid. Nonlinear Dyn. 98(3), 2097–2114 (2019)

    MATH  Google Scholar 

  58. Dehrouyeh-Semnani, A.M.: Nonlinear geometrically exact dynamics of fluid-conveying cantilevered hard magnetic soft pipe with uniform and nonuniform magnetizations. arXiv preprint arXiv:2203.14618 (2022)

  59. Bolotin, V.V.: The dynamic stability of elastic systems. Am. J. Phys. 33(9), 752–753 (1965)

    Google Scholar 

  60. Fu, Y., Zhong, J., Shao, X., Chen, Y.: Thermal postbuckling analysis of functionally graded tubes based on a refined beam model. Int. J. Mech. Sci. 96, 58–64 (2015)

    Google Scholar 

  61. Zuo, D., Ma, G., Cao, Y., Zhou, C., Luo, J.: The large amplitude response of functionally graded non-uniform and imperfect nanotube. Wave Random Complex (2021). https://doi.org/10.1080/17455030.2021.2014602

    Article  Google Scholar 

  62. Hu, X., Jin, Q., Fu, X.: Parametric resonance of shear deformable nanotubes: a novel higher-order model incorporating nonlinearity from both curvature and inertia. Eur. J. Mech. A-Solids 96, 104693 (2022)

    MathSciNet  MATH  Google Scholar 

  63. Dehrouyeh-Semnani, A.M., Zafari-Koloukhi, H., Dehdashti, E., Nikkhah-Bahrami, M.: A parametric study on nonlinear flow-induced dynamics of a fluid-conveying cantilevered pipe in post-flutter region from macro to micro scale. Int. J. Nonlinear Mech. 85, 207–225 (2016)

    Google Scholar 

  64. Lim, C.W., Zhang, G., Reddy, J.: A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids. 78, 298–313 (2015)

    MathSciNet  MATH  Google Scholar 

  65. Bahaadini, R., Saidi, A.R., Hosseini, M.: On dynamics of nanotubes conveying nanoflow. Int. J. Eng. Sci. 123, 181–196 (2018)

    MathSciNet  MATH  Google Scholar 

  66. Mahinzare, M., Mohammadi, K., Ghadiri, M.: A nonlocal strain gradient theory for vibration and flutter instability analysis in rotary SWCNT with conveying viscous fluid. Wave Random Complex 31(2), 305–330 (2021)

    MathSciNet  Google Scholar 

  67. Lu, L., Guo, X., Zhao, J.: On the mechanics of Kirchhoff and Mindlin plates incorporating surface energy. Int. J. Eng. Sci. 124, 24–40 (2018)

    MathSciNet  MATH  Google Scholar 

  68. Jin, Q., Ren, Y., Jiang, H., Li, L.: A higher-order size-dependent beam model for nonlinear mechanics of fluid-conveying FG nanotubes incorporating surface energy. Compos. Struct. 269, 114022 (2021)

    Google Scholar 

  69. Bahaadini, R., Hosseini, M.: Effects of nonlocal elasticity and slip condition on vibration and stability analysis of viscoelastic cantilever carbon nanotubes conveying fluid. Comput. Mater. Sci. 114, 151–159 (2016)

    Google Scholar 

  70. Beskok, A., Karniadakis, G.E.: Report: a model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Thermophys. Eng. 3(1), 43–77 (1999)

    Google Scholar 

  71. Rashidi, V., Mirdamadi, H.R., Shirani, E.: A novel model for vibrations of nanotubes conveying nanoflow. Comput. Mater. Sci. 51(1), 347–352 (2012)

    Google Scholar 

  72. Farajpour, A., Farokhi, H., Ghayesh, M.H.: A nonlinear viscoelastic model for NSGT nanotubes conveying fluid incorporating slip boundary conditions. J. Vib. Control 25(12), 1883–1894 (2019)

    MathSciNet  Google Scholar 

  73. Farajpour, A., Farokhi, H., Ghayesh, M.H., Hussain, S.: Nonlinear mechanics of nanotubes conveying fluid. Int. J. Eng. Sci. 133, 132–143 (2018)

    MathSciNet  MATH  Google Scholar 

  74. Atluri, S.: Nonlinear vibrations of a hinged beam including nonlinear inertia effects. J. Appl. Mech. 40(1), 121–126 (1973)

    MATH  Google Scholar 

  75. Shen, H.S., Li, C., Reddy, J.N.: Large amplitude vibration of FG-CNTRC laminated cylindrical shells with negative Poisson’s ratio. Comput. Method Appl. Mech. 360, 112727 (2020)

    MathSciNet  MATH  Google Scholar 

  76. Ren, Y., Li, L., Jin, Q., Nie, L., Peng, F.: Vibration and snap-through of fluid-conveying graphene reinforced composite pipes under low-velocity impact. AIAA J. 59(12), 5091–5105 (2021)

    Google Scholar 

  77. Barretta, R., de Sciarra, F.M.: Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams. Int. J. Eng. Sci. 130, 187–198 (2018)

    MathSciNet  MATH  Google Scholar 

  78. Li, L., Hu, Y., Li, X.: Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory. Int. J. Mech. Sci. 115, 135–144 (2016)

    Google Scholar 

  79. Dai, H.L., Abdelkefi, A., Wang, L.: Modeling and nonlinear dynamics of fluid-conveying risers under hybrid excitations. Int. J. Eng. Sci. 81, 1–14 (2014)

    MathSciNet  MATH  Google Scholar 

  80. Huang, J.L., Zhu, W.D.: Nonlinear dynamics of a high-dimensional model of a rotating Euler–Bernoulli beam under the gravity load. J. Appl. Mech.-T ASME. 81(10), 101007 (2014)

    Google Scholar 

  81. Xu, M.R., Xu, S.P., Guo, H.Y.: Determination of natural frequencies of fluid-conveying pipes using homotopy perturbation method. Comput. Math. Appl. 60(3), 520–527 (2010)

    MathSciNet  MATH  Google Scholar 

  82. Jin, Q.D., Yuan, F.G., Ren, Y.R.: Resonance interaction of flow-conveying nanotubes under forced vibration. Acta Mech. (2022) (in press)

  83. Aghamohammadi, M., Sorokin, V., Mace, B.: Dynamic analysis of the response of Duffing-type oscillators subject to interacting parametric and external excitations. Nonlinear Dyn. 107, 99–120 (2022)

    Google Scholar 

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Acknowledgements

The authors acknowledge the support by the National Natural Science Foundation of China (52172356) and Hunan Provincial Innovation Foundation for Postgraduate (CX20210384).

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Authors and Affiliations

  1. College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, Hunan, China

    Qiduo Jin & Yiru Ren

  2. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, Hunan, China

    Qiduo Jin & Yiru Ren

  3. Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, 27606, USA

    Fuh-Gwo Yuan

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QJ involved in theoretical modeling and computation, writing, discussion and analysis. YR took part in providing guidance, investigation, validation. FY took part in supervision, revising.

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Jin, Q., Ren, Y. & Yuan, FG. Combined resonance of pulsatile flow-transporting FG nanotubes under forced excitation with movable boundary. Nonlinear Dyn 111, 6157–6178 (2023). https://doi.org/10.1007/s11071-022-08148-1

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