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Exact solutions of vibration and postbuckling response of curved beam rested on nonlinear viscoelastic foundations

  • Nazira Mohamed (Department of Engineering Mathematics, Faculty of Engineering, Zagazig University) ;
  • Salwa A. Mohamed (Department of Engineering Mathematics, Faculty of Engineering, Zagazig University) ;
  • Mohamed A. Eltaher (Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University)
  • Received : 2024.01.25
  • Accepted : 2024.05.20
  • Published : 2024.03.25
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Abstract

This paper presents the exact solutions and closed forms for of nonlinear stability and vibration behaviors of straight and curved beams with nonlinear viscoelastic boundary conditions, for the first time. The mathematical formulations of the beam are expressed based on Euler-Bernoulli beam theory with the von Karman nonlinearity to include the mid-plane stretching. The classical boundary conditions are replaced by nonlinear viscoelastic boundary conditions on both sides, that are presented by three elements (i.e., linear spring, nonlinear spring, and nonlinear damper). The nonlinear integro-differential equation of buckling problem subjected to nonlinear nonhomogeneous boundary conditions is derived and exactly solved to compute nonlinear static response and critical buckling load. The vibration problem is converted to nonlinear eigenvalue problem and solved analytically to calculate the natural frequencies and to predict the corresponding mode shapes. Parametric studies are carried out to depict the effects of nonlinear boundary conditions and amplitude of initial curvature on nonlinear static response and vibration behaviors of curved beam. Numerical results show that the nonlinear boundary conditions have significant effects on the critical buckling load, nonlinear buckling response and natural frequencies of the curved beam. The proposed model can be exploited in analysis of macrosystem (airfoil, flappers and wings) and microsystem (MEMS, nanosensor and nanoactuators).

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References

  1. Adam, C., Ladurner, D. and Furtmuller, T. (2022), "Free and forced small flexural vibrations of slightly curved slender composite beams with interlayer slip", Thin Wall. Struct., 180, 109857. https://doi.org/10.1016/j.tws.2022.109857.
  2. Alessi, Y.A., Ali, I.A., Alazwari, M.A., Almitani, K.H., Abdelrahman, A. and Eltaher, M.A. (2023), "Dynamic analysis of piezoelectric perforated cantilever bimorph energy harvester via finite element analysis", Adv. Aircraft Spacecraft Sci., 10(2), 179-202. https://doi.org/10.12989/aas.2023.10.2.179.
  3. Almitani, K.H., Mohamed, N., Alazwari, M.A., Mohamed, S.A. and Eltaher, M.A. (2022), "Exact solution of nonlinear behaviors of imperfect bioinspired helicoidal composite beams resting on elastic foundations", Math., 10(6), 887. https://doi.org/10.3390/math10060887.
  4. Benguediab, S., Kebir, T., Kettaf, F.Z., Daikh, A.A, Tounsi, A., Benguediab, M. and Eltaher, M.A. (2023), "Thermomechanical behavior of Macro and Nano FGM sandwich plates", Adv. Aircraft Spacecraft Sci., 10(1) 83-106. https://doi.org/10.12989/aas.2023.10.1.083.
  5. Borjalilou, V., Taati, E. and Ahmadian, M.T. (2019), "Bending, buckling and free vibration of nonlocal FGcarbon nanotube-reinforced composite nanobeams: exact solutions", SN Appl. Sci., 1, 1-15. https://doi.org/10.1007/s42452-019-1359-6.
  6. Calim, F.F. (2012), "Forced vibration of curved beams on two-parameter elastic foundation", Appl. Math. Model., 36(3), 964-973. https://doi.org/10.1016/j.apm.2011.07.066.
  7. Chang, X., Zhou, J. and Li, Y. (2022), "Post-buckling characteristics of functionally graded fluid-conveying pipe with geometric defects on Pasternak foundation", Ocean Eng., 266, 113056. https://doi.org/10.1016/j.oceaneng.2022.113056.
  8. Chen, X. and Li, Y. (2018), "Size-dependent post-buckling behaviors of geometrically imperfect microbeams", Mech. Res. Commun., 88, 25-33. https://doi.org/10.1016/j.mechrescom.2017.12.005.
  9. Ding, H.X., She, G.L. and Zhang, Y.W. (2022), "Nonlinear buckling and resonances of functionally graded fluid-conveying pipes with initial geometric imperfection", Eur. Phys. J. Plus, 137(12), 1-18. https://doi.org/10.1140/epjp/s13360-022-03570-1.
  10. Ding, H., Lu, Z.Q. and Chen, L.Q. (2019), "Nonlinear isolation of transverse vibration of pre-pressure beams", J. Sound Vib., 442, 738-751. https://doi.org/10.1016/j.jsv.2018.11.028.
  11. Ecsedi, I. and Dluhi, K. (2005), "A linear model for the static and dynamic analysis of non-homogeneous curved beams", Appl. Math. Model., 29(12), 1211-1231. https://doi.org/10.1016/j.apm.2005.03.006.
  12. Eltaher, M.A. and Mohamed, N. (2020), "Nonlinear stability and vibration of imperfect CNTs by doublet mechanics", Appl. Math. Comput., 382, 125311. https://doi.org/10.1016/j.amc.2020.125311.
  13. Eltaher, M.A., Mohamed, N., Mohamed, S.A. and Seddek, L.F. (2019), "Periodic and nonperiodic modes of postbuckling and nonlinear vibration of beams attached to nonlinear foundations", Appl. Math. Model., 75, 414-445. https://doi.org/10.1016/j.apm.2019.05.026.
  14. Geng, X., Ding, H., Wei, K. and Chen, L. (2020), "Suppression of multiple modal resonances of a cantilever beam by an impact damper", Appl. Math. Mech., 41(3), 383-400. https://doi.org/10.1007/s10483-020-2588-9.
  15. Ghayesh, M.H., Farokhi, H. and Gholipour, A. (2017), "Vibration analysis of geometrically imperfect threelayered shear-deformable microbeams", Int. J. Mech. Sci., 122, 370-383. http://dx.doi.org/10.1016/j.ijmecsci.2017.01.001.
  16. 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(1), 51. https://doi.org/10.12989/aas.2023.10.1.051.
  17. Jin, Q., Hu, X., Ren, Y. and Jiang, H. (2020), "On static and dynamic snap-throughs of the imperfect postbuckled FG-GRC sandwich beams", J. Sound Vib., 489, 115684. https://doi.org/10.1016/j.jsv.2020.115684.
  18. Juhasz, Z. and Szekrenyes, A. (2020), "An analytical solution for buckling and vibration of delaminated composite spherical shells", Thin Wall. Struct., 148, 106563. https://doi.org/10.1016/j.tws.2019.106563.
  19. Lacarbonara, W. (1997), "A theoretical and experimental investigation of nonlinear vibrations of buckled beams", Doctoral Dissertation, Virginia Tech., USA.
  20. Lee, J.K. and Jeong, S. (2016), "Flexural and torsional free vibrations of horizontally curved beams on Pasternak foundations", Appl. Math. Model., 40(3), 2242-2256. https://doi.org/10.1016/j.apm.2015.09.024.
  21. Li, Z.M. and Qiao, P. (2014), "On an exact bending curvature model for nonlinear free vibration analysis shear deformable anisotropic laminated beams", Compos. Struct., 108, 243-258. http://dx.doi.org/10.1016/j.compstruct.2013.09.034.
  22. Ma, T.F. (2003), "Existence results and numerical solutions for a beam equation with nonlinear boundary conditions", Appl. Numer. Math., 47(2), 189-196. https://doi.org/10.1016/S0168-9274(03)00065-5.
  23. Ma, T.F. and Da Silva, J. (2004), "Iterative solutions for a beam equation with nonlinear boundary conditions of third order", Appl. Math. Comput., 159(1), 11-18. https://doi.org/10.1016/j.amc.2003.08.088.
  24. Mochida, Y. and Ilanko, S. (2016), "Condensation of independent variables in free vibration analysis of curved beams", Adv. Aircraft Spacecraft Sci., 3(1), 045. https://doi.org/10.12989/aas.2016.3.1.045.
  25. Mohamed, N., Eltaher, M.A., Mohamed, S.A. and Seddek, L.F. (2018), "Numerical analysis of nonlinear free and forced vibrations of buckled curved beams resting on nonlinear elastic foundations", Int. J. Nonlin. Mech., 101, 157-173. https://doi.org/10.1016/j.ijnonlinmec.2018.02.014.
  26. Mohamed, N., Mohamed, S.A. and Eltaher, M.A. (2022a), "Nonlinear static stability of imperfect bioinspired helicoidal composite beams", Math., 10(7), 1084. https://doi.org/10.3390/math10071084.
  27. Mohamed, S.A., Assie, A.E., Eltaher, M.A., Abo-bakr, R.M. and Mohamed, N. (2024a), "Nonlinear postbuckling and snap-through instability of movable simply supported BDFG porous plates rested on elastic foundations", Mech. Bas. Des. Struct. Mach., 1-28. https://doi.org/10.1080/15397734.2024.2328339.
  28. Mohamed, S.A., Eltaher, M.A., Mohamed, N. and Abo-bakr, R.M. (2024b), "Nonlinear postbuckling and snap-through instability of movable simply supported BDFG porous plates rested on elastic foundations", Mech. Bas. Des. Struct. Mach., 1-28. https://doi.org/10.1080/15397734.2024.2353321.
  29. Mohamed, S.A., Mohamed, N. and Eltaher, M.A. (2022b), "Snap-through instability of helicoidal composite imperfect beams surrounded by nonlinear elastic foundation", Ocean Eng., 263, 112171. https://doi.org/10.1016/j.oceaneng.2022.112171.
  30. Mohamed, N., Mohamed, S.A. and Eltaher, M.A. (2024c), "Nonlinear Forced Vibration of Curved Beam with Nonlinear Viscoelastic Ends", Int. J. Appl. Mech., 16(3), 2450031. https://doi.org/10.1142/S1758825124500315.
  31. Mohamed, S.A., Mohamed, N., Abo-bakr, R.M. and Eltaher, M.A. (2023), "Multi-objective optimization of snap-through instability of helicoidal composite imperfect beams using Bernstein polynomials method", Appl. Math. Model., 120, 301-329. https://doi.org/10.1016/j.apm.2023.03.034.
  32. Rezaiee-Pajand, M. and Kamali, F. (2021), "Exact solution for thermal-mechanical post-buckling of functionally graded micro-beams", CEAS Aeronaut. J., 12, 85-100. https://doi.org/10.1007/s13272-020-00480-9.
  33. Sedighi, H.M. and Shirazi, K.H. (2012), "A new approach to analytical solution of cantilever beam vibration with nonlinear boundary condition", J. Comput. Nonlin. Dyn., 7(3), 034502. https://doi.org/10.1115/1.4005924.
  34. Sedighi, H.M., Chan-Gizian, M. and Noghreha-Badi, A. (2014), "Dynamic pull-in instability of geometrically nonlinear actuated micro-beams based on the modified couple stress theory", Lat. Am. J. Solid. Struct., 11, 810-825.
  35. She, G.L. and Ding, H.X. (2023), "Nonlinear primary resonance analysis of initially stressed graphene platelet reinforced metal foams doubly curved shells with geometric imperfection", Acta Mechanica Sinica, 39(2), 522392. https://doi.org/10.1007/s10409-022-22392-x.
  36. Siam, O.A., Shanab, R.A., Eltaher, M.A. and Mohamed, N.A. (2023), "Free vibration analysis of nonlocal viscoelastic nanobeam with holes and elastic foundations by Navier analytical method", Adv. Aircraft Spacecraft Sci., 10(3), 257-279. https://doi.org/10.12989/aas.2023.10.3.257.
  37. Tekin, G., Ecer, S. and Kadioglu, F. (2023), "An alternative procedure for longitudinal vibration analysis of bars with arbitrary boundary conditions", J. Appl. Comput. Mech., 9(1), 294-301. https://doi.org/10.1016/j.ijmecsci.2021.106903.
  38. Wang, P., Cao, R., Deng, Y., Sun, Z., Luo, H. and Wu, N. (2023), "Vibration and resonance reliability analysis of non-uniform beam with randomly varying boundary conditions based on Kriging model", Struct., 50, 925-936. https://doi.org/10.1016/j.istruc.2023.02.050.
  39. Yang, Y.B., Liu, Y.H. and Xu, H. (2023), "Recovering mode shapes of curved bridges by a scanning vehicle", Int. J. Mech. Sci., 253, 108404. https://doi.org/10.1016/j.ijmecsci.2023.108404.
  40. Ye, S.Q., Mao, X.Y., Ding, H., Ji, J.C. and Chen, L.Q. (2020), "Nonlinear vibrations of a slightly curved beam with nonlinear boundary conditions", Int. J. Mech. Sci., 168, 105294. https://doi.org/10.1016/j.ijmecsci.2019.105294.
  41. Yuan, J.R. and Ding, H. (2022), "Dynamic model of curved pipe conveying fluid based on the absolute nodal coordinate formulation", Int. J. Mech. Sci., 232, 107625. https://doi.org/10.1016/j.ijmecsci.2022.107625.
  42. Yuan, J.R. and Ding, H. (2023), "An out-of-plane vibration model for in-plane curved pipes conveying fluid", Ocean Eng., 271, 113747. https://doi.org/10.1016/j.oceaneng.2023.113747.
  43. Zhai, Y.J., Ma, Z.S., Wang, B. and Ding, Q. (2023), "Dynamic characteristic analysis of beam structures with nonlinear elastic foundations and boundaries", Int. J. Nonlin. Mech., 153, 104409. https://doi.org/10.1016/j.ijnonlinmec.2023.104409.
  44. Zhao, Y., Du, J., Chen, Y. and Liu, Y. (2022), "Dynamic behavior analysis of the axially loaded beam with the nonlinear support and elastic boundary constraints", Chin. J. Theor. Appl. Mech., 54(9), 2529-2542. https://doi/10.6052/0459-1879-22-088.