Nonlinear Functional Analysis and Applications

  • 제27권2호
  • /
  • Pages.405-432
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  • 2022
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  • 1229-1595(pISSN)
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  • 2466-0973(eISSN)
경남대학교 수학교육과 (Kyungnam University, Department of Mathematics Eduaction)
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EQUATIONS OF MOTION FOR CRACKED BEAMS AND SHALLOW ARCHES

  • Gutman, Semion (Department of Mathematics, University of Oklahoma) ;
  • Ha, Junhong (School of Liberal Arts, Korea University of Technology and Education) ;
  • Shon, Sudeok (Department of Architectural Engineering, Korea University of Technology and Education)
  • 투고 : 2021.12.01
  • 심사 : 2022.01.17
  • 발행 : 2022.06.08
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초록

Cracks in beams and shallow arches are modeled by massless rotational springs. First, we introduce a specially designed linear operator that "absorbs" the boundary conditions at the cracks. Then the equations of motion are derived from the first principles using the Extended Hamilton's Principle, accounting for non-conservative forces. The variational formulation of the equations is stated in terms of the subdifferentials of the bending and axial potential energies. The equations are given in their abstract (weak), as well as in classical forms.

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과제정보

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2020R1I1A1A01065032). And this paper was supported by Education and Research promotion program of the KOREATECH in 2021.

참고문헌

  1. J.M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42(1) (1973), 61-90. https://doi.org/10.1016/0022-247x(73)90121-2
  2. J.M. Ball, Stability theory for an extensible beam, J. Diff. Equ., 14(3) (1973), 399-418. https://doi.org/10.1016/0022-0396(73)90056-9
  3. V. Barbu, Nonlinear differential equations of monotone type in Banach spaces, Springer-Verlag, New York, 2010.
  4. S. Caddemi and I. Calio, Exact closed-form solution for the vibration modes of the EulerBernoulli beam with multiple open cracks, J. Sound Vibration, 327(3) (2009), 473-489. https://doi.org/10.1016/j.jsv.2009.07.008
  5. S. Caddemi and A. Morassi, Multi-cracked Euler-Bernoulli beams: Mathematical modeling and exact solutions, Inter. J. Solids Structures, 50(6) (2013), 944-956. https://doi.org/10.1016/j.ijsolstr.2012.11.018
  6. F. Cannizzaro and A. Greco and S. Caddemi and I. Calio, Closed form solutions of a multi-cracked circular arch under static loads, Inter. J. Solids Structures, 121 (2017), 944-956.
  7. M.N. Cerri and G.C. Ruta, Detection of localised damage in plane circular arches by frequency data, Journal of Sound and Vibration, 270(1) (2004), 39-59. https://doi.org/10.1016/S0022-460X(03)00482-6
  8. T.G. Chondros and A.D. Dimarogonas and J. Yao, A continuous cracked beam vibration theory, J. Sound Vibration,215(1) (1998), 17-34. https://doi.org/10.1006/jsvi.1998.1640
  9. S. Christides and A.D.S. Barr, One-dimensional theory of cracked Bernoulli-Euler beams, International Journal of Mechanical Sciences, 26(11) (1984), 639-648. https://doi.org/10.1016/0020-7403(84)90017-1
  10. A.D. Dimarogonas, Vibration of cracked structures: a state of the art review, Eng. Fracture Mechanics, 55(5) (1996), 831-857. https://doi.org/10.1016/0013-7944(94)00175-8
  11. E. Emmrich and M. Thalhammer, A class of integro-differential equations incorporating nonlinear and nonlocal damping with applications in nonlinear elastodynamics: Existence via time discretization, Nonlinearity, 24(9) (2011), 2523-2546. https://doi.org/10.1088/0951-7715/24/9/008
  12. S. Gutman and J. Ha, Shallow arches with weak and strong damping, J. Kor. Math. Soc., 54 (2017), 945-966. https://doi.org/10.4134/JKMS.j160317
  13. S. Gutman and J. Ha, Uniform attractor of shallow arch motion under moving points load, J. Math. Anal. Appl., 464(1) (2018), 557-579. https://doi.org/10.1016/j.jmaa.2018.04.025
  14. S. Gutman, J. Ha and S. Lee, Parameter identification for weakly damped shallow arches, J. Math. Anal. Appl., 403(1) (2013), 297-313. https://doi.org/10.1016/j.jmaa.2013.02.047
  15. J. Ha and S. Gutman and S. Shon, Variational setting for cracked beams and shallow arches, to appear in Arch. Appl. Mech., (2022).
  16. S.M. Han, H. Benaroya and T. Wei, Dynamics of transversely vibrating beams using four engineering theories, J. Sound Vibration, 225(5) (1999), 935-988. https://doi.org/10.1006/jsvi.1999.2257
  17. J. Kim, G. F. Dargush and Y. Ju, Extended framework of Hamilton's principle for continuum dynamics, Inter. J. Solids Structures, 50(20) (2013), 3418-3429. https://doi.org/10.1016/j.ijsolstr.2013.06.015
  18. H.P. Lin, S.C. Chang and J.D. Wu, Beam vibrations with an arbitrary number of cracks, J. Sound Vibration, 258(5) (2002), 987-999. https://doi.org/10.1006/jsvi.2002.5184
  19. W.M. Ostachowicz and M. Krawczuk, Analysis of the effect of cracks on the natural frequencies of a cantilever beam, J. Sound Vibration 150(2) (1991), 191-201. https://doi.org/10.1016/0022-460X(91)90615-Q
  20. E.I. Shifrin and R. Ruotolo, Natural frequencies of a beam with an arbitrary number of cracks, J. Sound Vibration, 222(3) (1999), 409-423. https://doi.org/10.1006/jsvi.1998.2083
  21. R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer, New York, 1997.
  22. S. Woinowsky-Krieger, The effect of axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. https://doi.org/10.1115/1.4010053