Journal of the Korea Academia-Industrial cooperation Society (한국산학기술학회논문지)

The Korea Academia-Industrial cooperation Society (한국산학기술학회)
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In-Plane Inextensional and Extensional Vibration Analysis of Curved Beams Using DQM

미분구적법(DQM)을 이용한 곡선보의 내평면 비신장 및 신장 진동해석

  • Kang, Ki-jun (Department of Mechanical Engineering, Hoseo University)
  • 강기준 (호서대학교 공과대학 기계공학부)
  • Received : 2015.03.25
  • Accepted : 2015.11.06
  • Published : 2015.11.30
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Abstract

One of the efficient procedures for the solution of partial differential equations is the method of differential quadrature. This method has been applied to a large number of cases to circumvent the difficulties of the complex algorithms of programming for the computer, as well as excessive use of storage due to conditions of complex geometry and loading. In-plane vibrations of curved beams with inextensibility and extensibility of the arch axis are analyzed by the differential quadrature method (DQM). Fundamental frequencies are calculated for the member with various end conditions and opening angles. The results are compared with exact experimental and numerical results by other methods for cases in which they are available. The DQM gives good accuracy even when only a limited number of grid points is used, and new results according to diverse variation are also suggested.

편미분방정식 해를 위한 효율적인 방법 중의 하나는 미분구적법이다. 이방법은 복잡한 구조 및 하중에 따른 컴퓨터 용량의 과도한 사용뿐만 아니라, 컴퓨터 프로그래밍의 복합알고리즘 해석상의 어려움 피하기 위해 많은 분야에 적용되어왔다. 아크축의 비신장 및 신장을 고려한 곡선보의 내평면 진동을, 미분구적법 (DQM)을 이용하여 해석하였다. 다양한 경계조건과 열림각에 따른 진동수을 계산하였다. DQM의 해석결과는, 비교 가능한 정확한 수학적 해법을 다른 수치해석결과와 비교하였다. DQM은 적은 격자점을 사용하고도 정확한 해석을 보여주었고, 다양한 변화에 따른 새로운 결과를 제시하였다.

Keywords

References

  1. R. Hoppe, "The Bending Vibration of a Circular Ring", Crelle's J. Math., Vol. 73, pp. 158-170, 1871. DOI: http://dx.doi.org/10.1515/crll.1871.73.158
  2. A. E. H. Love, A Treatise of the Mathematical Theory of Elasticity, 4th Ed. Dover, New York, 1944.
  3. H. Lamb, "On the Flexure and Vibrations of a Curved Bar", Proceedings of the London Mathematical Society, Vol. 19, pp. 365-376, 1888.
  4. J. P. Den Hartog, "The Lowest Natural Frequency of Circular Arcs", Philosophical Magazine, Series 7, Vol. 5, pp. 400-408, 1928. DOI: http://dx.doi.org/10.1080/14786440208564480
  5. E. Volterra and J. D. Morell, "Lowest Natural Frequency of Elastic Arc for Vibrations outside the Plane of Initial Curvature", J. Appl. Math., Vol. 28, pp. 624-627, 1961. DOI: http://dx.doi.org/10.1115/1.3641794
  6. R. R. Archer, "Small Vibration of Thin Incomplete Circular Rings", Int. J. Mech. Sci., Vol. 1, pp. 45-56, 1960. DOI: http://dx.doi.org/10.1016/0020-7403(60)90029-1
  7. F. C. Nelson, "In-Plane Vibration of a Simply Supported Circular Ring Segment" Int. J. Mech. Sci., Vol. 4, pp. 517-527, 1962. DOI: http://dx.doi.org/10.1016/S0020-7403(62)80013-7
  8. N. M. Auciello and M. A. De Rosa, "Free Vibrations of Circular Arches", J. Sound Vibr., Vol. 176, pp. 443-458, 1994. DOI: http://dx.doi.org/10.1006/jsvi.1994.1388
  9. U. Ojalvo, "Coupled Twisting-Bending Vibrations of Incomplete Elastic Rings", Int. J. Mech. Sci., Vol. 4, pp. 53-72, 1962. DOI: http://dx.doi.org/10.1016/0020-7403(62)90006-1
  10. L. C. Rodgers and W. H. Warner, "Dynamic Stability of Out-of-Plane Motion of Curved Elastic Rods", J. Appl. Math., Vol. 24, pp. 36-43, 1973. DOI: http://dx.doi.org/10.1137/0124005
  11. R. E. Bellman and J. Casti, "Differential Quadrature and Long-Term Integration", J. Math. Anal. Applic., Vol. 34, pp. 235-238, 1971. DOI: http://dx.doi.org/10.1016/0022-247X(71)90110-7
  12. A. S. Veletsos, W. J. Austin, C. A. L. Pereira, and S. J. Wung, "Free In-plane Vibration of Circular Arches", Proceedings ASCE, Journal of the Engineering Mechanics Division, Vol. 98, pp. 311-339, 1972.
  13. W. Flugge, Stresses in Shells, Springer- Verlag, Berlin, 1960. DOI: http://dx.doi.org/10.1007/978-3-662-29731-5
  14. S. K. Jang, C. W. Bert, and A. G. Striz, "Application of Differential Quadrature to Static Analysis of Structural Components", Int. J. Numer. Mech. Engng, Vol. 28, pp. 561-577, 1989. DOI: http://dx.doi.org/10.1002/nme.1620280306
  15. K. Kang and J. Han, "Analysis of a Curved beam Using Classical and Shear Deformable Beam Theories", Int. J. KSME., Vol. 12, pp. 244-256, 1998. https://doi.org/10.1007/BF02947169
  16. K. Kang and Y. Kim, "In-Plane Buckling Analysis of Curved Beams Using DQM", J. KAIS., Vol. 13, pp. 2858-2864, 2012. DOI: http://dx.doi.org/10.5762/kais.2012.13.7.2858
  17. K. Kang and Y. Kim, "In-Plane Vibration Analysis of Asymmetric Curved Beams Using DQM", J. KAIS., Vol. 11, pp. 2734-2740, 2010. DOI: http://dx.doi.org/10.5762/kais.2010.11.8.2734
  18. K. Kang and C. Park, "In-Plane Buckling Analysis of Asymmetric Curved Beams Using DQM", J. KAIS., Vol.141, pp .4706-4712, 2013.