Steel and Composite Structures

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A new solution for dynamic response of FG nonlocal beam under moving harmonic load

  • Hosseini, S.A.H. (Department of Industrial, Mechanical and Aerospace Engineering, Buein Zahra Technical University) ;
  • Rahmani, O. (Smart Structures and New Advanced Materials Laboratory, Department of Mechanical Engineering, University of Zanjan) ;
  • Bayat, S. (Smart Structures and New Advanced Materials Laboratory, Department of Mechanical Engineering, University of Zanjan)
  • Received : 2020.03.05
  • Accepted : 2022.04.04
  • Published : 2022.04.25
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Abstract

A Closed-form solution for dynamic response of a functionally graded (FG) nonlocal nanobeam due to action of moving harmonic load is presented in this paper. Due to analyzing in small scale, a nonlocal elasticity theory is utilized. The governing equation and boundary conditions are derived based on the Euler-Bernoulli beam theory and Hamilton's principle. The material properties vary through the thickness direction. The harmonic moving load is modeled by Delta function and the FG nanobeam is simply supported. Using the Laplace transform the dynamic response is obtained. The effect of important parameters such as excitation frequency, the velocity of the moving load, the power index law of FG material and the nonlocal parameter is analyzed. To validate, the results were compared with previous literature, which showed an excellent agreement.

Keywords

References

  1. Ansari, R. and Sahmani, S. (2012), "Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models", Commun. Nonlinear Sci. Numer. Simulat., 17(4), 1965-1979. https://doi.org/10.1016/j.cnsns.2011.08.043.
  2. Apuzzo, A., Barretta, R., Faghidian, S.A., Luciano, R. and Marotti de Sciarra, F. (2018), "Free vibrations of elastic beams by modified nonlocal strain gradient theory", Int. J. Eng. Sci.. 133 99-108. https://doi.org/10.1016/j.ijengsci.2018.09.002.
  3. Apuzzo, A., Barretta, R., Faghidian, S.A., Luciano, R. and Marotti de Sciarra, F. (2019), "Nonlocal strain gradient exact solutions for functionally graded inflected nano-beams", Compos. Part B: Eng., 164 667-674. https://doi.org/10.1016/j.compositesb.2018.12.112.
  4. Apuzzo, A., Barretta, R., Luciano, R., de Sciarra, F.M. and Penna, R. (2017), "Free vibrations of Bernoulli-Euler nano-beams by the stress-driven nonlocal integral model", Compos. Part B: Eng., 123 105-111. https://doi.org/10.1016/j.compositesb.2017.03.057.
  5. Aydogdu, M. and Elishakoff, I. (2014), "On the vibration of nanorods restrained by a linear spring in-span", Mech. Res. Commun., 57 90-96. https://doi.org/10.1016/j.mechrescom.2014.03.003.
  6. Bahrami, A. (2017), "Free vibration, wave power transmission and reflection in multi-cracked nanorods", Compos. Part B: Eng., 127 53-62. https://doi.org/10.1016/j.compositesb.2017.06.024.
  7. Bahrami, A., Zargaripoor, A., Shiri, H. and Khosravi, N. (2019), "Size-dependent free vibration of axially functionally graded tapered nanorods having nonlinear spring constraint with a tip nanoparticle", J. Vib. Control, 2769-2783. https://doi.org/10.1177%2F1077546319870921. https://doi.org/10.1177%2F1077546319870921
  8. Barretta, R. and Marotti de Sciarra, F. (2018), "Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams", Int. J. Eng. Sci., 130 187-198. https://doi.org/10.1016/j.ijengsci.2018.05.009.
  9. Barretta, R., Canadija, M., Feo, L., Luciano, R., Marotti de Sciarra, F. and Penna, R. (2018), "Exact solutions of inflected functionally graded nano-beams in integral elasticity", Compos. Part B: Eng., 142 273-286. https://doi.org/10.1016/j.compositesb.2017.12.022.
  10. Barretta, R., Canadija, M., Luciano, R. and de Sciarra, F.M. (2018), "Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams", Int. J. Eng. Sci., 126 53-67. https://doi.org/10.1016/j.compositesb.2017.12.022.
  11. Barretta, R., Caporale, A., Faghidian, S.A., Luciano, R., Marotti de Sciarra, F. and Medaglia, C.M. (2019), "A stress-driven local-nonlocal mixture model for Timoshenko nano-beams", Compos. Part B: Eng., 164 590-598. https://doi.org/10.1016/j.compositesb.2019.01.012.
  12. Barretta, R., Diaco, M., Feo, L., Luciano, R., de Sciarra, F.M. and Penna, R. (2018), "Stress-driven integral elastic theory for torsion of nano-beams", Mech. Res. Commun., 87 35-41. https://doi.org/10.1016/j.mechrescom.2017.11.004.
  13. Barretta, R., Fabbrocino, F., Luciano, R. and Sciarra, F.M.D. (2018), "Closed-form solutions in stress-driven two-phase integral elasticity for bending of functionally graded nano-beams", Physica E: Low-dimensional Syst. Nanostruct., 97 13-30. https://doi.org/10.1016/j.physe.2017.09.026.
  14. Barretta, R., Faghidian, S.A. and Luciano, R. (2019), "Longitudinal vibrations of nano-rods by stress-driven integral elasticity", Mech. Adv. Mater. Struct., 26(15), 1307-1315. https://doi.org/10.1080/15376494.2018.1432806.
  15. Barretta, R., Faghidian, S.A. and Marotti de Sciarra, F. (2019), "Stress-driven nonlocal integral elasticity for axisymmetric nano-plates", Int. J. Eng. Sci., 136 38-52. https://doi.org/10.1016/j.ijengsci.2019.01.003.
  16. Barretta, R., Faghidian, S.A., Marotti de Sciarra, F., Penna, R. and Pinnola, F.P. (2020), "On torsion of nonlocal Lam strain gradient FG elastic beams", Compos. Struct., 233 111550. https://doi.org/10.1016/j.compstruct.2019.111550.
  17. Barretta, R., Luciano, R., de Sciarra, F.M. and Ruta, G. (2018), "Stress-driven nonlocal integral model for Timoshenko elastic nano-beams", Europ. J. Mech.-A/Solids. 72 275-286. https://doi.org/10.1016/j.euromechsol.2018.04.012.
  18. Barretta, R., Luciano, R., Marotti de Sciarra, F. and Ruta, G. (2018), "Stress-driven nonlocal integral model for Timoshenko elastic nano-beams", Europ. J. Mech. - A/Solids. 72 275-286. https://doi.org/10.1016/j.euromechsol.2018.04.012.
  19. Bastanfar, M., Hosseini, S.A., Sourki, R. and Khosravi, F. (2019), "Flexoelectric and surface effects on a cracked piezoelectric nanobeam: Analytical resonant frequency response", Archve. Mech. Eng., 66(4), 417. http://dx.doi.org/10.24425/ame.2019.131355.
  20. Ebrahimi, F. and Salari, E. (2015), "Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment", Acta Astronautica. 113, 29-50. https://doi.org/10.1016/j.actaastro.2015.03.031.
  21. Ebrahimi, F. and Salari, E. (2015), "Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments", Compos. Struct., 128 363-380. https://doi.org/10.1016/j.compstruct.2015.03.023.
  22. Ebrahimi, F., Ghadiri, M., Salari, E., Hoseini, S.A.H. and Shaghaghi, G.R. (2015), "Application of the differential transformation method for nonlocal vibration analysis of functionally graded nanobeams", J. Mech. Sci. Technol., 29(3), 1207-1215. https://doi.org/10.1007/s12206-015-0234-7.
  23. Eltaher, M., Emam, S.A. and Mahmoud, F. (2012), "Free vibration analysis of functionally graded size-dependent nanobeams", Appl. Mathem. Comput., 218(14), 7406-7420. https://doi.org/10.1016/j.amc.2011.12.090.
  24. Eltaher, M., Emam, S.A. and Mahmoud, F. (2013), "Static and stability analysis of nonlocal functionally graded nanobeams", Compos. Struct., 96, 82-88. https://doi.org/10.1016/j.compstruct.2012.09.030.
  25. Eltaher, M., Khairy, A., Sadoun, A. and Omar, F.A. (2014), "Static and buckling analysis of functionally graded Timoshenko nanobeams", Appl. Mathem. Comput., 229, 283-295. https://doi.org/10.1016/j.amc.2013.12.072.
  26. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803.
  27. Faghidian, S.A. (2018), "Integro-differential nonlocal theory of elasticity", Int. J. Eng. Sci., 129 96-110. https://doi.org/10.1016/j.ijengsci.2018.04.007.
  28. Faghidian, S.A. (2018), "On non-linear flexure of beams based on non-local elasticity theory", Int. J. Eng. Sci., 124 49-63. https://doi.org/10.1016/j.ijengsci.2017.12.002.
  29. Fang, B., Zhen, Y.X., Zhang, C.P. and Tang, Y. (2013), "Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory", Appl. Mathem. Modelling. 37(3), 1096-1107. https://doi.org/10.1016/j.apm.2012.03.032.
  30. Hamidi, B.A., Hosseini, S.A., Hassannejad, R. and Khosravi, F. (2019), "An exact solution on gold microbeam with thermoelastic damping via generalized Green-Naghdi and modified couple stress theories", J. Thermal Stresses. 1-18. https://doi.org/10.1080/01495739.2019.1666694
  31. Hamidi, B.A., Hosseini, S.A., Hassannejad, R. and Khosravi, F. (2020), "Theoretical analysis of thermoelastic damping of silver nanobeam resonators based on Green-Naghdi via nonlocal elasticity with surface energy effects", Europ. Phys. J. Plus. 135(1), 35. https://doi.org/10.1140/epjp/s13360-019-00037-8.
  32. Hosseini-Hashemi, S., Nahas, I., Fakher, M. and Nazemnezhad, R. (2014), "Surface effects on free vibration of piezoelectric functionally graded nanobeams using nonlocal elasticity", Acta Mechanica. 225(6), 1555-1564. https://doi.org/10.1007/s00707-013-1014-z.
  33. Hosseini, S. and Rahmani, O. (2017), "Exact solution for axial and transverse dynamic response of functionally graded nanobeam under moving constant load based on nonlocal elasticity theory", Meccanica. 52(6), 1441-1457. https://doi.org/10.1007/s11012-016-0491-2.
  34. Hosseini, S.A. and Khosravi, F. (2020), "Exact solution for dynamic response of size dependent torsional vibration of CNT subjected to linear and harmonic loadings", Adv. Nano Res., 8(1), 25-36. https://doi.org/10.12989/anr.2020.8.1.025.
  35. Hosseini, S.A., Khosravi, F. and Ghadiri, M. (2019), "Moving axial load on dynamic response of single-walled carbon nanotubes using classical, Rayleigh and Bishop rod models based on Eringen's theory", J. Vib. Control. 1077546319890170. https://doi.org/10.1177/1077546319890170.
  36. Hosseini, S.A., Khosravi, F. and Ghadiri, M. (2020), "Effect of external moving torque on dynamic stability of carbon nanotube", J. Nano Res., 61, 118-135. https://doi.org/10.4028/www.scientific.net/JNanoR.61.118.
  37. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature. 354(6348), 56-58. https://doi.org/10.1038/354056a0.
  38. Khosravi, F., Hosseini, S.A. and Hamidi, B.A. (2020), "On torsional vibrations of triangular nanowire", Thin-Wall. Struct., 148 106591. https://doi.org/10.1016/j.tws.2019.106591.
  39. Khosravi, F., Hosseini, S.A. and Norouzi, H. (2020), "Exponential and harmonic forced torsional vibration of single-walled carbon nanotube in an elastic medium", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 0954406220903341.
  40. 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", Europ. Phys. J. Plus. 135(2), 183. https://doi.org/10.1140/epjp/s13360-020-00207-z.
  41. Kiani, K. (2010), "Longitudinal and transverse vibration of a single-walled carbon nanotube subjected to a moving nanoparticle accounting for both nonlocal and inertial effects", Physica E: Low-dimensional Syst. Nanostruct., 42(9), 2391-2401. https://doi.org/10.1016/j.physe.2010.05.021.
  42. Kiani, K. (2011), "Small-scale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theory", J. Sound Vib., 330(20), 4896-4914. https://doi.org/10.1016/j.jsv.2011.03.033.
  43. Kiani, K. (2014), "Longitudinal and transverse instabilities of moving nanoscale beam-like structures made of functionally graded materials", Compos. Struct., 107 610-619. https://doi.org/10.1016/j.compstruct.2013.07.035.
  44. Li, C., Li, S., Yao, L. and Zhu, Z. (2015), "Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models", Appl. Mathem. Modelling. 39(15), 4570-4585. https://doi.org/10.1016/j.apm.2015.01.013.
  45. Murmu, T. and Pradhan, S. (2009), "Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory", Comput. Mater. Sci., 46(4), 854-859. https://doi.org/10.1016/j.commatsci.2009.04.019.
  46. 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.
  47. Niknam, H. and Aghdam, M.M. (2015), "A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation", Compos. Struct., 119, 452-462. https://doi.org/10.1016/j.compstruct.2014.09.023.
  48. Ozgur Yayli, M. (2018), "An efficient solution method for the longitudinal vibration of nanorods with arbitrary boundary conditions via a hardening nonlocal approach", J. Vib. Control. 24(11), 2230-2246. https://doi.org/10.1177%2F1077546316684042. https://doi.org/10.1177%2F1077546316684042
  49. Pirmohammadi, A.A., Pourseifi, M., Rahmani, O. and Hoseini, S.A.H. (2014), "Modeling and active vibration suppression of a single-walled carbon nanotube subjected to a moving harmonic load based on a nonlocal elasticity theory", Appl. Phys. A. 117(3), 1547-1555. https://doi.org/10.1007/s00339-014-8592-z.
  50. Pourseifi, M., Rahmani, O. and Hoseini, S.A.H. (2015), "Active vibration control of nanotube structures under a moving nanoparticle based on the nonlocal continuum theories", Meccanica, 50(5), 1351-1369. https://doi.org/10.1007/s11012-014-0096-6.
  51. Rahmani, O. and Pedram, O. (2014), "Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory", Int. J. Eng. Sci., 77 55-70. https://doi.org/10.1016/j.ijengsci.2013.12.003.
  52. Rahmani, O., Norouzi, S., Golmohammadi, H. and Hosseini, S. (2017), "Dynamic response of a double, single-walled carbon nanotube under a moving nanoparticle based on modified nonlocal elasticity theory considering surface effects", Mech. Adv. Mater. Struct., 24(15), 1274-1291. https://doi.org/10.1080/15376494.2016.1227504.
  53. Rahmani, O., Shokrnia, M., Golmohammadi, H. and Hosseini, S. (2018), "Dynamic response of a single-walled carbon nanotube under a moving harmonic load by considering modified nonlocal elasticity theory", Europ. Phys. J. Plus. 133(2), 1-13. https://doi.org/10.1140/epjp/i2018-11868-4.
  54. Romano, G. and Barretta, R. (2017), "Nonlocal elasticity in nanobeams: the stress-driven integral model", Int. J. Eng. Sci., 115, 14-27. https://doi.org/10.1016/j.ijengsci.2017.03.002.
  55. Romano, G. and Barretta, R. (2017), "Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams", Compos. Part B. 114, 184-188. https://doi.org/10.1016/j.compositesb.2017.01.008.
  56. Romano, G., Luciano, R., Barretta, R. and Diaco, M. (2018), "Nonlocal integral elasticity in nanostructures, mixtures, boundary effects and limit behaviours", Continuum Mech. Thermodyn., 30(3), 641-655. https://doi.org/10.1007/s00161-018-0631-0.
  57. Shafiei, N., Ghadiri, M., Makvandi, H. and Hosseini, S.A. (2017), "Vibration analysis of Nano-Rotor's Blade applying Eringen nonlocal elasticity and generalized differential quadrature method", Appl. Mathema. Modelling. 43 191-206. https://doi.org/10.1016/j.apm.2016.10.061.
  58. Shahsavari, D., Karami, B., Janghorban, M. and Li, L. (2017), "Dynamic characteristics of viscoelastic nanoplates under moving load embedded within visco-Pasternak substrate and hygrothermal environment", Mater. Res. Express. 4(8), 085013. https://doi.org/10.1088/2053-1591/aa7d89
  59. Simsek, M. (2010), "Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory", Physica E: Low-Dimens. Syst. Nanostruct., 43(1), 182-191. https://doi.org/10.1016/j.physe.2010.07.003.
  60. Simsek, M. (2011), "Nonlocal effects in the forced vibration of an elastically connected double-carbon nanotube system under a moving nanoparticle", Comput. Mater. Sci., 50(7), 2112-2123. https://doi.org/10.1016/j.commatsci.2011.02.017.
  61. Uymaz, B. (2013), "Forced vibration analysis of functionally graded beams using nonlocal elasticity", Compos. Struct., 105 227-239. https://doi.org/10.1016/j.compstruct.2013.05.006.
  62. Wang, C., Zhang, Y. and He, X. (2007), "Vibration of nonlocal Timoshenko beams", Nanotechnology. 18(10), 105401. https://doi.org/10.1088/0957-4484/18/10/105401
  63. Wang, C., Zhang, Y., Ramesh, S.S. and Kitipornchai, S. (2006), "Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory", J. Phys. D: Appl. Phys., 39(17), 3904. https://doi.org/10.1088/0022-3727/39/17/029
  64. Zarepour, M., Hosseini, S. and Akbarzadeh, A. (2019), "Geometrically nonlinear analysis of Timoshenko piezoelectric nanobeams with flexoelectricity effect based on Eringen's differential model", Appl. Mathem. Modelling, 69, 563-582. https://doi.org/10.1016/j.apm.2019.01.001.