Advances in aircraft and spacecraft science

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Parametric resonance of graphene platelet-reinforced metal foam beams under external excitation and boundary constraints

  • W.B. Shan (Hunan Electrical College of Technology) ;
  • H. Li (Hunan Electrical College of Technology) ;
  • Q.M. Peng (Hunan Electrical College of Technology) ;
  • N.N. Zhang (Changsha Environmental Protection College)
  • Received : 2024.07.18
  • Accepted : 2025.08.25
  • Published : 2025.09.25
⟨ Previous Next ⟩

Abstract

This paper investigates the parametric resonance of graphene-platelet reinforced metal foam (GPLRMF) beams under three typical boundary conditions. The Halpin-Tsai model and rule of mixture are employed to characterize the GPLRMF material properties, while the governing equations are established based on Euler-Bernoulli beam theory. Numerical results demonstrate that increased external excitation leads to significant amplification of vibration amplitudes. The modified variable amplitude method is adopted to ensure accurate prediction of system response across the entire frequency range, with validation against existing literature. Detailed parametric studies examine the effects of: (1) foam distributions (including Foam-I pattern), (2) porosity coefficients, (3) GPL dispersion patterns (Type-A to Type-X), and (4) GPL weight fractions. Furthermore, the influences of boundary constraints, thermal environments, external stimuli, and damping ratios on parametric resonance are systematically analyzed. Key findings indicate that Foam-I distribution with Type-A GPL arrangement exhibits negligible dynamic response, suggesting superior anti-vibration capacity. Both clamped boundaries and reduced temperature are shown to effectively shift resonance positions while enhancing beam stiffness.

Keywords

Acknowledgement

We gratefully acknowledge the financial support from the following funding projects: Hunan Electrical College of Technology, 411101.Xiangtan, PR China, and Changsha Environmental Protection College, Changsha, 410004, People's Republic of China.

References

  1. Abouelregal, A.E. (2018), "Response of thermoelastic microbeams to a periodic external transverse excitation based on MCS theory", Microsyst. Technol., 24(4), 1925-1933. https://doi.org/10.1007/s00542-017-3589-0.
  2. Abouelregal, A.E. (2020), "Size-dependent thermoelastic initially stressed micro-beam due to a varying temperature in the light of the modified couple stress theory", Appl. Math. Mech., 41(12), 1805-1820. https://doi.org/10.1007/s10483-020-2676-5.
  3. Abouelregal, A.E. (2021), "Thermoelastic fractional derivative model for exciting viscoelastic microbeam resting on Winkler foundation", J. Vib. Control, 27(17-18), 2123-2135. https://doi.org/10.1177/1077546320956528.
  4. Abouelregal, A.E. (2022), "Modeling and analysis of a thermoviscoelastic rotating micro-scale beam under laser heat models of", Thin Wall. Struct., 174, 109150. https://doi.org/10.1016/j.tws.2022.109150.
  5. Abouelregal, A.E. (2024), "Effect of non-local modified couple stress theory on the responses of axially moving thermoelastic nano-beams", ZAMM-Zeitschrift fur Angewandte Mathematik und Mechanik, 104(4), 1-22. https://doi.org/10.1002/zamm.202200233.
  6. Abouelregal, A.E. and Tiwari, R. (2022), "The thermoelastic vibration of nano-sized rotating beams with variable thermal properties under axial load via memory-dependent heat conduction", Meccanica, 57(8), 2001-2025. https://doi.org/10.1007/s11012-022-01543-3.
  7. Abouelregal, A.E., Akgoz, B. and Civalek, O. (2022), "Nonlocal thermoelastic vibration of a solid medium subjected to a pulsed heat flux via Caputo-Fabrizio fractional derivative heat conduction", Appl. Phys. A, 128(8), 660. https://doi.org/10.1007/s00339-022-05786-5.
  8. Aghamohammadi, M., Sorokin, V. and Mace, B. (2019), "On the response attainable in nonlinear parametrically excited systems", Appl. Phys. Lett., 115(15), 154102. https://doi.org/10.1063/1.5120434.
  9. Aghamohammadi, M., Sorokin, V. and Mace, B. (2023), "Nonlinear dynamics of parametrically excited cantilever beams with a tip mass considering nonlinear inertia and Duffing-type nonlinearity", Nonlin. Dyn., 111(8), 7251-7269. https://doi.org/10.1007/s11071-023-08236-w.
  10. Akbas, S.D., Dastjerdi, S., Akgoez, B. and Civalek, O. (2022), "Dynamic analysis of functionally graded porous microbeams under moving load", Transp. Porous Media, 142(1), 209-227. https://doi.org/10.1007/s11242-021-01686-z.
  11. Al-Furjan, M.S.H., Habibi, M., Jung, D.W., Chen, G.J., Safarpour, M. and Safarpour, H. (2021), "Chaotic responses and nonlinear dynamics of the graphene nanoplatelets reinforced doubly-curved panel", Eur. J. Mech. A-Solid., 85, 104091. https://doi.org/10.1016/j.euromechsol.2020.104091.
  12. Alhassan, Y. and Abouelregal, A.E. (2025), "Impact of microscopic interactions and non-Local dynamics on rotating nanobeam structures under external moving loads", Contin. Mech. Thermodyn., 37(4), 62. https://doi.org/10.1007/s00161-025-01390-z.
  13. Alibakhshi, A., Dastjerdi, S., Akgoz, B. and Civalek, O. (2022), "Parametric vibration of a dielectric elastomer microbeam resonator based on a hyperelastic cosserat continuum model", Compos. Struct., 287, 115386. https://doi.org/10.1016/j.compstruct.2022.115386.
  14. Alimoradzadeh, M. and Akbas, S.D. (2023), "Nonlinear vibration analysis of carbon nanotube-reinforced composite beams resting on nonlinear viscoelastic foundation", Geomech. Eng., 32(2), 125-135. https://doi.org/10.12989/gae.2023.32.2.125.
  15. Arvin, H., Arena, A. and Lacarbonara, W. (2020), "Nonlinear vibration analysis of rotating beams undergoing parametric instability: Lagging-axial motion", Mech. Syst. Signal Pr., 144, 106892. https://doi.org/10.1016/j.ymssp.2020.106892.
  16. Ashraf, M.A., Liu, Z.L., Zhang, D.Q. and Pham, B.T. (2022), "Effects of elastic foundation on the large-amplitude vibration analysis of functionally graded GPL-RC annular sector plates", Eng. Comput., 38(1), 325-345. https://doi.org/10.1007/s00366-020-01068-x.
  17. Carboni, B., Catarci, S. and Lacarbonara, W. (2022), "Parametric resonances of nonlinear piezoelectric beams exploiting in-plane actuation", Mech. Syst. Signal Pr., 163, 108119. https://doi.org/10.1016/j.ymssp.2021.108119.
  18. Chai, Y.Y., Li, F.M., Song, Z.G. and Zhang, C.Z. (2020), "Analysis and active control of nonlinear vibration of composite lattice sandwich plates", Nonlin. Dyn., 102(4), 2179-2203. https://doi.org/10.1007/s11071-020-06059-7.
  19. Chen, W., Wang, L. and Dai, H.L. (2020), "Nonlinear free vibration of hyperelastic beams based on neo-hookean model", Int. J. Struct. Stab. Dy., 20(1), 2050015. https://doi.org/10.1142/S0219455420500157.
  20. Cheng, Y.H. and She, G.L. (2025a), "Nonlinear transient response of rotating graphene-reinforced metal foam beams with cracks", Int. J. Struct. Stab. Dyn., 2640003. https://doi.org/10.1142/S0219455426400031.
  21. Cheng, Y.H. and She, G.L. (2025b), "Nonlinear combined resonance of graphene platelet-reinforced metal foam beam", Steel Compos. Struct., 56(3), 265-276. https://doi.org/10.12989/scs.2025.56.3.265.
  22. Cheng, Y.H., She, G.L. and Eltaher, M.A. (2025), "Nonlinear internal resonance of graphene-reinforced metal foam plates with initial geometric imperfection under non-uniform temperature field", Aerosp. Sci. Technol., 165, 110491. https://doi.org/10.1016/j.ast.2025.110491.
  23. El Khouddar, Y., Adri, A., Outassafte, O., Rifai, S. and Benamar, R. (2021), "An analytical approach to geometrically nonlinear free and forced vibration of piezoelectric functional gradient beams resting on elastic foundations in thermal environments", Mech. Adv. Mater. Struct., 30(1), 131-143. https://doi.org/10.1080/15376494.2021.2009601.
  24. Fan, Y.H., She, G.L. and Li, C. (2025), "Nonlinear transient response analysis of revolution doubly curved shells", Arch. Civil Mech. Eng., 25, 145. https://doi.org/10.1007/s43452-025-01187-6.
  25. Foroutan, K. and Ahmadi, H. (2022), "Nonlinear parametric vibration of imperfect SSMFG cylindrical shells in thermal environment including internal and subharmonic resonances", Mech. Adv. Mater. Struct., 29(24), 3499-3522. https://doi.org/10.1080/15376494.2021.1904526.
  26. Garg, A. and Dwivedy, S.K. (2020), "Dynamic analysis of piezoelectric energy harvester under combination parametric and internal resonance: a theoretical and experimental study", Nonlin. Dyn., 101(4), 2107-2129. https://doi.org/10.1007/s11071-020-05931-w.
  27. Gu, X.J., Zhang, W. and Zhang, Y.F. (2021), "Nonlinear vibrations of rotating pretwisted composite blade reinforced by functionally graded graphene platelets under combined aerodynamic load and airflow in tip clearance", Nonlin. Dyn., 105(2), 1503-1532. https://doi.org/10.1007/s11071-021-06681-z.
  28. Hamed, Y.S. and Alkhathami, H.K. (2021), "On the nonlinear vibrations and stability analysis of a plate-cavity system with a nonlinear restoring force", IEEE Access, 9, 20423-20439. https://doi.org/10.1109/ACCESS.2021.3054517.
  29. Hasan, A.M.Z. and Rahman, M.S. (2022), "Multi-level residue harmonic balance method for nonlinear vibration of the beam", J. Low Freq. Noise Vib. Act. Control, 41(1), 278-291. https://doi.org/10.1177/14613484211038403.
  30. Hosseini, S.A.H., Rahmani, O., Refaeinejad, V., Golmohammadi, H. and Montazeripour, M. (2023), "Free vibration of deep and shallow curved FG nanobeam based on nonlocal elasticity", Adv. Aircraft Spacecraft Sci., 10, 51-65. https://doi.org/10.12989/aas.2023.10.1.051.
  31. Hu, Y.D., Li, Z., Du, G.J. and Wang, Y.N. (2018), "Magneto-elastic combination resonance of rotating circular plate with varying speed under alternating load", Int. J. Struct. Stab. Dyn., 18(3), 1850032. https://doi.org/10.1142/S0219455418500323.
  32. Hu, Y.D., Rong, Y.T. and Li, J. (2018), "Primary parametric resonance of an axially accelerating beam subjected to static loads", J. Theor. Appl. Mech., 56(3), 815-828. https://doi.org/10.15632/jtam-pl.56.3.815.
  33. Jahangiri, R., Rezaee, M. and Manafi, H. (2022), "Nonlinear and chaotic vibrations of FG double curved sandwich shallow shells resting on visco-elastic nonlinear Hetenyi foundation under combined resonances", Compos. Struct., 295, 115721. https://doi.org/10.1016/j.compstruct.2022.115721.
  34. Le, C.I. and Nguyen, D.K. (2023), "Nonlinear vibration of three-phase bidirectional functionally graded sandwich beams with influence of homogenization scheme and partial foundation support", Compos. Struct., 307, 116649. https://doi.org/10.1016/j.compstruct.2022.116649.
  35. Li, X. (2021), "Parametric resonances of rotating composite laminated nonlinear cylindrical shells under periodic axial loads and hygrothermal environment", Compos. Struct., 255, 112887. https://doi.org/10.1016/j.compstruct.2020.112887.
  36. Li, X.Q., Song, M.T., Yang, J. and Kitipornchai, S. (2019), "Primary and secondary resonances of functionally graded graphene platelet-reinforced nanocomposite beams", Nonlin. Dyn., 95(3), 1807-1826. https://doi.org/10.1007/s11071-018-4660-9.
  37. Li, Y.P. and She, G.L. (2025), "Nonlinear dynamic response of graphene platelets reinforced cylindrical shells under moving loads considering initial geometric imperfection", Eng. Struct., 323(A), 119241. https://doi.org/10.1016/j.engstruct.2024.119241.
  38. Li, Z., Hu, Y.D. and Li, J. (2017), "Magnetoelastic principal parametric resonance of a rotating electroconductive circular plate", Shock Vib., 2017, 5196847. https://doi.org/10.1155/2017/5196847.
  39. Li. H.Y., Zhang, S.M., Li, J. and Wang, X.B. (2020), "Parametric resonance of axially accelerating unidirectional plates partially immersed in fluid", J. Comput. Nonlin. Dyn., 15(10), 101002. https://doi.org/10.1115/1.4047483.
  40. Liu, T., Zhang, W. and Wang, J.F. (2017), "Nonlinear dynamics of composite laminated circular cylindrical shell clamped along a generatrix and with membranes at both ends", Nonlin. Dyn., 90(2), 1393-1417. https://doi.org/10.1007/s11071-017-3734-4.
  41. Ma, Z.S., She, G.L. and Li, C. (2025), "Nonlinear thermal flutter analysis of graphene platelets reinforced metal foam arbitrary quadrilateral plates", Arch. Civil Mech. Eng., 25, 203. https://doi.org/10.1007/s43452-025-01262-y.
  42. Mbong, T.L.M.D., Siewe, M.S. and Tchawoua, C. (2018), "Controllable parametric excitation effect on linear and nonlinear vibrational resonances in the dynamics of a buckled beam", Commun. Nonlin. Sci., 54, 377-388. https://doi.org/10.1016/j.cnsns.2017.06.019.
  43. Mirjavadi, S.S., Forsat, M., Hamouda, A. and Barati, M.R. (2019), "Dynamic response of functionally graded graphene nanoplatelet reinforced shells with porosity distributions under transverse dynamic loads", Mater. Res. Expr., 6(7), 075045. https://doi.org/10.1088/2053-1591/ab1552.
  44. Mohamed, N., Mohamed, S.A. and Eltaher, M.A. (2024), "Exact solutions of vibration and postbuckling response of curved beam rested on nonlinear viscoelastic foundations", Adv. Aircraft Spacecraft Sci., 11, 55-81. https://doi.org/10.12989/aas.2024.11.1.055.
  45. Rincon-Casado, A., Gonzalez-Carbajal, J., Garcia-Vallejo, D. and Dominguez, J. (2021), "Analytical and numerical study of the influence of different support types in the nonlinear vibrations of beams", Eur. J. Mech. A-Solid., 85, 104113. https://doi.org/10.1016/j.euromechsol.2020.104113.
  46. Sahoo, B. (2020), "Nonlinear dynamics of a viscoelastic beam traveling with pulsating speed, variable axial tension under two-frequency parametric excitations and internal resonance", Nonlin. Dyn., 99(2), 945-979. https://doi.org/10.1007/s11071-019-05264-3.
  47. Sahoo, B. (2022), "Multi-scale analysis of a moving beam under parametric and auto-parametric resonances", J. Brazil. Soc. Mech. Sci., 44(1), 20. https://doi.org/10.1007/s40430-021-03303-y.
  48. Sahoo, B., Panda, L.N. and Pohit, G. (2017), "Stability, bifurcation and chaos of a traveling viscoelastic beam tuned to 3: 1 internal resonance and subjected to parametric excitation", Int. J. Bifurcat. Chaos, 27(2), 1750017. https://doi.org/10.1142/S0218127417500171.
  49. Sahoo, P.K. and Chatterjee, S. (2021), "High-frequency vibrational control of principal parametric resonance of a nonlinear cantilever beam: Theory and experiment", J. Sound. Vib., 505, 116138. https://doi.org/10.1016/j.jsv.2021.116138.
  50. Shao, Y.F. and Ding, H. (2023), "Evaluation of gravity effects on the vibration of fluid-conveying pipes", Int. J. Mech. Sci., 248, 108230. https://doi.org/10.1016/j.ijmecsci.2023.108230.
  51. She, G.L., Gan, L.L. and Xu, J.Q. (2025), "Nonlinear snap-buckling of graphene platelet reinforced metal foams doubly curved shells with geometric imperfection", Geomech. Eng., 40(3), 183-191. https://doi.org/10.12989/gae.2025.40.3.183.
  52. Sheng, G.G. and Wang, X. (2018), "Nonlinear vibration of FG beams subjected to parametric and external excitations", Eur. J. Mech. A-Solid., 71, 224-234. https://doi.org/10.1016/j.euromechsol.2018.04.003.
  53. Teng, M.W. and Wang, Y.Q. (2021), "Nonlinear forced vibration of simply supported functionally graded porous nanocomposite thin plates reinforced with graphene platelets", Thin. Wall. Struct., 164, 107799. https://doi.org/10.1016/j.tws.2021.107799.
  54. Wang, L., Yang, J. and Li, Y.H. (2021), "Nonlinear vibration of a deploying laminated Rayleigh beam with a spinning motion in hygrothermal environment", Eng. Comput., 37(4), 3825-3841. https://doi.org/10.1007/s00366-020-01035-6.
  55. Wang, T.Q., Wang, R.H. and Ma, N.J. (2019), "Nonlinear vibration of a stiffened plate considering the existence of initial stresses", KSCE J. Civil Eng., 23(5), 2303-2312. https://doi.org/10.1007/s12205-019-1387-1.
  56. Wang, W.Q., Ye, C. and Zu, J.W. (2019), "Nonlinear vibration of metal foam cylindrical shells reinforced with graphene platelets", Aerosp. Sci. Technol., 85, 359-370. https://doi.org/10.1016/j.ast.2018.12.022.
  57. Wang, Y., Jing, X., Dai, H. and Li, F. (2019b), "Subharmonics and ultra-subharmonics of a bio-inspired nonlinear isolation", Int. J. Mech. Sci., 152, 167-184. https://doi.org/10.1016/j.ijmecsci.2018.12.054.
  58. Wang, Y., Li, F. and Shu, H. (2019a), "Nonlocal nonlinear chaotic and homoclinic analysis of double layered forced viscoelastic nanoplates", Mech. Syst. Signal Pr., 122, 537-554. https://doi.org/10.1016/j.ymssp.2018.12.041.
  59. Wang, Y., Liu, J., Li, J., Jing, X. and Wiercigroch, M. (2025), "Intrinsic mechanisms of vibrational mode jumping in sandwich panels", J. Sound Vib., 618(Part B), 119247. https://doi.org/10.1016/j.jsv.2025.119247.
  60. Wang, Y.W. and Zhang, W. (2022), "On the thermal buckling and postbuckling responses of temperature-dependent graphene platelets reinforced porous nanocomposite beams", Compos. Struct., 296, 115880. https://doi.org/10.1016/j.compstruct.2022.115880.
  61. Wang, Y.W., Xie, K., Fu, T.R. and Shi, C.L. (2019), "Vibration response of a functionally graded graphene nanoplatelet reinforced composite beam under two successive moving masses", Compos. Struct., 209, 928-939. https://doi.org/10.1016/j.compstruct.2018.11.014.
  62. Xie, K., Wang, Y.W. and Fu, T.R. (2020), "Nonlinear vibration analysis of third-order shear deformable functionally graded beams by a new method based on direct numerical integration technique", Int. J. Mech. Mater. Des., 16(4), 839-855. https://doi.org/10.1007/s10999-020-09493-y.
  63. Xiong, X., Wang, Y., Li, J. and Li, F. (2023), "Internal resonance analysis of bio-inspired X-shaped structure with nonlinear vibration absorber", Mech. Syst. Signal Pr., 185, 109809. https://doi.org/10.1016/j.ymssp.2022.109809.
  64. Xu, H., Wang, Y.Q. and Zhang, Y.F. (2021), "Free vibration of functionally graded graphene platelet-reinforced porous beams with spinning movement via differential transformation method", Arch. Appl. Mech., 91(12), 4817-4834. https://doi.org/10.1007/s00419-021-02036-7.
  65. Xu, J.Q., Li, Y.P. and She, G.L. (2025), "Nonlinear low-velocity impact response of GPLRMF doubly curved shells in thermal environment", Comput. Concrete, 35(3), 219-229. https://doi.org/10.12989/cac.2025.35.3.219.
  66. Yang, C.M., Jin, G.Y., Zhang, J.H., Ye, T.G. and Liu, Z.G. (2021), "A unified Fourier spectral method for nonlinear free vibration analysis of the laminated composite and sandwich beams with arbitrary restrained ends", J. Brazil. Soc. Mech. Sci. Eng., 43(9), 432. https://doi.org/10.1007/s40430-021-03150-x.
  67. Yang, F.L., Wang, Y.Q. and Liu, Y.F. (2022), "Low-velocity impact response of axially moving functionally graded graphene platelet reinforced metal foam plates", Aerosp. Sci. Technol., 123, 107496. https://doi.org/10.1016/j.ast.2022.107496.
  68. Yang, J. and Chen, Y. (2008), "Free vibration and buckling analyses of functionally graded beams with edge cracks", Compos. Struct., 83(1), 48-60. https://doi.org/10.1016/j.compstruct.2007.03.006.
  69. Zhang, Y.F., Zhang, W. and Yao, Z.G. (2018), "Analysis on nonlinear vibrations near internal resonances of a composite laminated piezoelectric rectangular plate", Eng. Struct., 173, 89-106. https://doi.org/10.1016/j.engstruct.2018.04.100.
  70. Zhang, Z., Gao, Z.T., Fang, B. and Zhang, Y.W. (2022), "Vibration suppression of a geometrically nonlinear beam with boundary inertial nonlinear energy sinks", Nonlin. Dyn., 109(3), 1259-1275. https://doi.org/10.1007/s11071-022-07490-8.
  71. Zhao, B. and She, G.L. (2025), "Vibration analysis of graphene reinforced metal foam coupled plates under arbitrary boundary and coupled conditions", Eng. Struct., 343, 121143. https://doi.org/10.1016/j.engstruct.2025.121143.
  72. Zhao, Y.H., Du, J.T., Chen, Y.L. and Liu, Y. (2023), "Nonlinear dynamic behavior of a generally restrained pre-pressure beam with a partial non-uniform foundation of nonlinear stiffness", Int. J. Struct. Stab. Dyn., 23(03), 2350028. https://doi.org/10.1142/S0219455423500281.
  73. Zhou, Y.X., Zhang, Y.M. and Yao, G. (2021), "Nonlinear forced vibration analysis of a rotating three-dimensional tapered cantilever beam", J. Vib. Control, 27(15-16), 1879-1892. https://doi.org/10.1177/1077546320949716.
  74. Zhu, B., Dong, Y.H. and Li, Y.H. (2018), "Nonlinear dynamics of a viscoelastic sandwich beam with parametric excitations and internal resonance", Nonlin. Dyn., 94(4), 2575-2612. https://doi.org/10.1007/s11071-0184511-8.
  75. Zhu, B., Xu, Q., Li,M and Li, Y.H. (2020), "Nonlinear free and forced vibrations of porous functionally graded pipes conveying fluid and resting on nonlinear elastic foundation", Compos. Struct., 252, 112672. https://doi.org/10.1016/j.compstruct.2020.112672.