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Nonlocal fully intrinsic equations for free vibration of Euler–Bernoulli beams with constitutive boundary conditions

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Abstract

Fully intrinsic beam theory has presented a relatively new set of equations for modeling of one-dimensional structures. Although this method can be utilized as a useful tool for modeling beam-like structures, it is scale independent and cannot be accurately employed when the size of a structure becomes comparable with its internal characteristic length. In this paper, we develop nonlocal fully intrinsic equations for accommodating the small-scale effect and overcoming this limitation. The original fully intrinsic equations are thus reformulated using the nonlocal differential constitutive relations of Eringen. The so-called constitutive boundary conditions are considered. Here the generalized differential quadrature method is used for numerically solving the nonlocal fully intrinsic equations. Furthermore, applicability of the present formulations for free vibration of both uniform and nonuniform Euler–Bernoulli beams is discussed. Moreover, it is shown that the present formulation is accurate and useful for predicting the mechanical behavior of micro- and nanobeams.

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References

  1. Romero, I.: A comparison of finite elements for nonlinear beams: the absolute nodal coordinate and geometrically exact formulations. Multibody Syst. Dyn. 20, 51–68 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauchau, O.A., Han, S.L.: Three-dimensional beam theory for flexible multibody dynamics. J. Comput. Nonlinear Dyn. 9, 041011 (2014)

    Article  Google Scholar 

  3. Demoures, F., Gay-Balmaz, F., Leyendecker, S., Ober-Blöbaum, S., Ratiu, T.S., Weinand, Y.: Discrete variational Lie group formulation of geometrically exact beam dynamics. Numer. Math. 130, 73–123 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Han, S.L., Bauchau, O.A.: Nonlinear three-dimensional beam theory for flexible multibody dynamics. Multibody Syst. Dyn. 34, 211–242 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bauchau, O.A., Kang, N.K.: A multibody formulation for helicopter structural dynamic analysis. J. Am. Helicopter Soc. 38, 3–14 (1993)

    Article  Google Scholar 

  6. Hodges, D.H.: A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams. Int. J. Solids Struct. 26, 1253–1273 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hodges, D.H.: Geometrically exact, intrinsic theory for dynamics of curved and twisted anisotropic beams. AIAA J. 41, 1131–1137 (2003)

    Article  Google Scholar 

  8. Hodges, D.H., Patil, M.J.: Correlation of geometrically-exact beam theory with the Princeton data. J. Am. Helicopter Soc. 49, 357–360 (2004)

    Article  Google Scholar 

  9. Chang, C.S., Hodges, D.H.: Vibration characteristics of curved beams. J. Mech. Mater. Struct. 4, 675–692 (2009)

    Article  Google Scholar 

  10. Chang, C.S., Hodges, D.H.: Stability studies for curved beams. J. Mech. Mater. Struct. 4, 1257–1270 (2009)

    Article  Google Scholar 

  11. Sotoudeh, Z., Hodges, D.H.: Modeling beams with various boundary conditions using fully intrinsic equations. J. Appl. Mech. 78, 031010 (2011)

    Article  Google Scholar 

  12. Patil, M.J., Hodges, D.H.: Flight dynamics of highly flexible flying wings. J. Aircraft 43, 1790–1799 (2006)

    Article  Google Scholar 

  13. Chang, C.S., Hodges, D.H.: Parametric studies on ground vibration test modeling for highly flexible aircraft. J. Aircraft 44, 2049–2059 (2007)

    Article  Google Scholar 

  14. Chang, C.S., Hodges, D.H., Patil, M.J.: Flight dynamics of highly flexible aircraft. J. Aircraft 45, 538–545 (2008)

    Article  Google Scholar 

  15. Sotoudeh, Z., Hodges, D.H., Chang, C.S.: Validation studies for aeroelastic trim and stability of highly flexible aircraft. J. Aircraft 47, 1240–1247 (2010)

    Article  Google Scholar 

  16. Sotoudeh, Z., Hodges, D.H.: Incremental method for structural analysis of joined-wing aircraft. J. Aircraft 48, 1588–1601 (2011)

    Article  Google Scholar 

  17. Sotoudeh, Z., Hodges, D.H.: Structural dynamics analysis of rotating blades using fully intrinsic equations, Part I: formulations. J. Am. Helicopter Soc. 58, 032004 (2013)

    Google Scholar 

  18. Sotoudeh, Z., Hodges, D.H.: Structural dynamics analysis of rotating blades using fully intrinsic equations, Part II: verification of dual load path configurations. J. Am. Helicopter Soc. 58, 032003 (2013)

    Google Scholar 

  19. Mardanpour, P., Hodges, D.H., Neuhart, R., Graybeal, N.: Effect of engine placement on aeroelastic trim and stability of flying-wing aircraft. J. Aircraft 50, 1716–1725 (2013)

    Article  Google Scholar 

  20. Mardanpour, P., Hodges, D.H.: Passive morphing of flying wing aircraft: Z-shaped configuration. J. Fluids Struct. 44, 17–30 (2014)

    Article  Google Scholar 

  21. Richards, P.W., Yao, Y., Mardanpour, P., Herd, R.A., Hodges, D.H.: Effect of inertial and constitutive properties on body-freedom flutter of a flying wing. J. Aircraft 53, 756–767 (2016)

    Article  Google Scholar 

  22. Patil, M.J., Althoff, M.: Energy-consistent, Galerkin approach for the nonlinear dynamics of beams using intrinsic equations. J. Vib. Control 17, 1748–1758 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Palacios, R.: Nonlinear normal modes in an intrinsic theory of anisotropic beams. J. Sound Vib. 330, 1772–1792 (2011)

    Article  Google Scholar 

  24. Hesse, H., Palacios, R.: Consistent structural linearisation in flexible-body dynamics with large rigid-body motion. Comput. Struct. 110–111, 1–14 (2012)

    Article  Google Scholar 

  25. Hesse, H., Palacios, R.: Reduced-order aeroelastic models for dynamics of maneuvering flexible aircraft. AIAA J. 52, 1717–1732 (2014)

    Article  Google Scholar 

  26. Wang, Y., Palacios, R., Wynn, A.: A method for normal-mode-based model reduction in nonlinear dynamics of slender structures. Comput. Struct. 159, 26–40 (2015)

    Article  Google Scholar 

  27. Wang, Y., Wynn, A., Palacios, R.: Nonlinear modal aeroservoelastic analysis framework for flexible aircraft. AIAA J. 54, 3075–3090 (2016)

    Article  Google Scholar 

  28. Khaneh Masjedi, P., Ovesy, H.R.: Chebyshev collocation method for static intrinsic equations of geometrically exact beams. Int. J. Solids Struct. 54, 183–191 (2015)

    Article  MATH  Google Scholar 

  29. Khaneh Masjedi, P., Ovesy, H.R.: Large deflection analysis of geometrically exact spatial beams under conservative and nonconservative loads using intrinsic equations. Acta Mech. 226, 1689–1706 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Amoozgar, M.R., Shahverdi, H.: Analysis of nonlinear fully intrinsic equations of geometrically exact beams using generalized differential quadrature method. Acta Mech. 227, 1265–1277 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Amoozgar, M.R., Shahverdi, H.: Dynamic instability of beams under tip follower forces using geometrically exact, fully intrinsic equations. Lat. Am. J. Solids and Struct. 13, 3022–3038 (2016)

    Article  MATH  Google Scholar 

  32. Mardanpour, P., Izadpanahiy, E., Rastkar, S., Fazelzadeh, S.A., Hodges, D.H.: Geometrically-exact, fully intrinsic analysis of initially twisted beams under distributed follower forces. AIAA J. 56, 836–848 (2018)

    Article  Google Scholar 

  33. Arash, B., Wang, Q.: A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput. Mater. Sci. 51, 303–313 (2012)

    Article  Google Scholar 

  34. Rafii-Tabar, H., Ghavanloo, E., Fazelzadeh, S.A.: Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Phys. Rep. 638, 1–97 (2016)

    Article  MathSciNet  Google Scholar 

  35. Eltaher, M.A., Khater, M.E., Emam, S.A.: A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl. Math. Model. 40, 4109–4128 (2016)

    Article  MathSciNet  Google Scholar 

  36. Askari, H., Younesian, D., Esmailzadeh, E., Cveticanin, L.: Nonlocal effect in carbon nanotube resonators: a comprehensive review. Adv. Mech. Eng. 9, 1–24 (2017)

    Article  Google Scholar 

  37. Behera, L., Chakraverty, S.: Recent researches on nonlocal elasticity theory in the vibration of carbon nanotubes using beam models: a review. Arch. Comput. Methods Eng. 24, 481–494 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Taghizadeh, M., Ovesy, H.R., Ghannadpour, S.A.M.: Beam buckling analysis by nonlocal integral elasticity finite element method. Int. J. Struct. Stab. Dyn. 16, 1550015 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Romano, G., Barretta, R., Diaco, M., de Sciarra, F.M.: Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. Int. J. Mech. Sci. 121, 151–156 (2017)

    Article  Google Scholar 

  40. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)

    MATH  Google Scholar 

  41. Du, H., Liew, K.M., Lim, M.K.: Generalized differential quadrature method for buckling analysis. J. Eng. Mech. 122, 95–100 (1996)

    Article  Google Scholar 

  42. Hodges, D.H.: Nonlinear Composite Beam Theory. American Institute of Aeronautics and Astronautics, Virginia (2006)

    Book  Google Scholar 

  43. Sotoudeh, Z.: Nonlinear static and dynamic analysis of beam structures using fully intrinsic equations. Ph.D. thesis, Georgia Institute of Technology (2011)

  44. Lazar, M., Maugin, G.A., Aifantis, E.C.: On a theory of nonlocal elasticity of bi-Helmholtz type and some applications. Int. J. Solids Struct. 43, 1404–1421 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Benvenuti, E., Simone, A.: One-dimensional nonlocal and gradient elasticity: closed-form solution and size effect. Mech. Res. Commun. 48, 46–51 (2013)

    Article  Google Scholar 

  46. Romano, G., Barretta, R., Diaco, M.: On nonlocal integral models for elastic nano-beams. Int. J. Mech. Sci. 131–132, 490–499 (2017)

    Article  Google Scholar 

  47. Wu, C.P., Lee, C.Y.: Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness. Int. J. Mech. Sci. 43, 1853–1869 (2001)

    Article  MATH  Google Scholar 

  48. Pradhan, S.C., Kumar, A.: Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method. Comput. Mater. Sci. 50, 239–245 (2010)

    Article  Google Scholar 

  49. Abrate, S.: Vibration of non-uniform rods and beams. J. Sound Vib. 185, 703–716 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wang, C.M., Zhang, Y.Y., He, X.Q.: Vibration of nonlocal Timoshenko beam. Nanotechnology 18, 105401 (2007)

    Article  Google Scholar 

  51. Hosseini Hashemi, S., Bakhshi Khaniki, H.: Analytical solution for free vibration of variable cross-section nonlocal nanobeam. Int. J. Eng. Trans. B Appl. 29, 688–696 (2016)

    Google Scholar 

  52. Eltaher, M.A., Alshorbagy, A.E., Mahmoud, F.F.: Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Appl. Math. Model. 37, 4787–4797 (2013)

    Article  MathSciNet  Google Scholar 

  53. Nguyen, N.T., Kim, N.I., Lee, J.: Mixed finite element analysis of nonlocal Euler–Bernoulli nanobeams. Finite Elem. Anal. Des. 106, 65–72 (2015)

    Article  Google Scholar 

  54. Wang, Y.B., Zhu, X.W., Dai, H.H.: Exact solutions for the static bending of Euler–Bernoulli beams using Eringen’s two-phase local/nonlocal model. AIP Adv. 6, 085114 (2016)

    Article  Google Scholar 

  55. Xu, X.J., Deng, Z.C., Zhang, K., Xu, W.: Observations of the softening phenomena in the nonlocal cantilever beams. Compos. Struct. 145, 43–57 (2016)

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Mechanical Engineering, Shiraz University, Shiraz, 71963-16548, Iran

    Mohammad Tashakorian, Esmaeal Ghavanloo & S. Ahmad Fazelzadeh

  2. School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0150, USA

    Dewey H. Hodges

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  1. Mohammad Tashakorian
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  2. Esmaeal Ghavanloo
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  3. S. Ahmad Fazelzadeh
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  4. Dewey H. Hodges
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Correspondence to S. Ahmad Fazelzadeh.

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Tashakorian, M., Ghavanloo, E., Fazelzadeh, S.A. et al. Nonlocal fully intrinsic equations for free vibration of Euler–Bernoulli beams with constitutive boundary conditions. Acta Mech 229, 3279–3292 (2018). https://doi.org/10.1007/s00707-018-2164-9

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  • DOI: https://doi.org/10.1007/s00707-018-2164-9

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