대한기계학회논문집A (Transactions of the Korean Society of Mechanical Engineers A)

  • 제41권1호
  • /
  • Pages.31-40
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  • 2017
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  • 1226-4873(pISSN)
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  • 2288-5226(eISSN)
대한기계학회 (The Korean Society of Mechanical Engineers)
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절대절점좌표계에서 선형 강성행렬을 활용한 2차원 보의 무차원 해석

Non-Dimensional Analysis of a Two-Dimensional Beam Using Linear Stiffness Matrix in Absolute Nodal Coordinate Formulation

  • 투고 : 2016.07.18
  • 심사 : 2016.09.22
  • 발행 : 2017.01.01
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초록

1990년대 중반에 개발된 절대절점좌표는 탄성체 동역학 해석에 활용되고 있다. 운동방정식을 유도하는 과정에서 변위장을 이루는 다항식의 차수가 증가하면 필연적으로 자유도가 증가하게 되고, 이는 해석 시간의 증가로 이어진다. 따라서 본 연구에서는 차원 운동방정식을 무차원 운동방정식으로 전환함으로써 해석 시간을 단축시키고자 하였다. 위치 벡터를 이루는 형상 함수는 무차원으로, 절점 좌표는 길이 차원으로 정리한 후 무차원화하는 변수를 통해 무차원 질량행렬, 무차원 선형 강성행렬 및 무차원 보존력을 유도하였다. 무차원 운동방정식의 검증과 효율성은 정적 처짐에 대한 정해가 존재하는 외팔보 및 단진자 예제를 통해 제시하였다.

Absolute nodal coordinate formulation was developed in the mid-1990s, and is used in the flexible dynamic analysis. In the process of deriving the equation of motion, if the order of polynomial referring to the displacement field increases, then the degrees of freedom increase, as well as the analysis time increases. Therefore, in this study, the primary objective was to reduce the analysis time by transforming the dimensional equation of motion to a non-dimensional equation of motion. After the shape function was rearranged to be non-dimensional and the nodal coordinate was rearranged to be in length dimension, the non-dimensional mass matrix, stiffness matrix, and conservative force was derived from the non-dimensional variables. The verification and efficiency of this non-dimensional equation of motion was performed using two examples; cantilever beam which has the exact solution about static deflection and flexible pendulum.

키워드

참고문헌

  1. Shabana, A. A. and Hussein, H. A., 1996, "An Absolute Nodal Coordination for the Large Rotation and Large Deformation Analysis of Flexible Bodies," Technical Report, No.MBS96-1-UIC.
  2. Shabana, A. A., 1997, "Definition of the Slopes and the Finite Element Absolute Nodal Coordinate Formulation," Multibody System Dynamics, Vol. 1, pp. 339-348. https://doi.org/10.1023/A:1009740800463
  3. Shabana, A. A., Hussein, H. A. and Escalona, J. L., 1998, "Application of the Absolute Nodal Coordinate Formulation to Large Rotation and Large Deformation Problems," Journal of Mechanical Design, Vol. 120, No. 2, pp. 188-195. https://doi.org/10.1115/1.2826958
  4. Shabana. A. A., 2005, Dynamics of Multibody Systems, Cambridge University Press, New York, Third edition.
  5. Dmitrochenko, O., Yoo, W. S. and Pogorelov, D. Y., 2006, "Helicoseir as Shape of a Rotating String (I): 2D Theory and Simulation Using ANCF," Multibody System Dynamics, Vol. 15, pp. 135-158. https://doi.org/10.1007/s11044-005-9002-2
  6. Dmitrochenko, O., Yoo, W. S. and Pogorelov, D. Y., 2006, "Helicoseir as Shape of a Rotating String (II): 3D Theory and Simulation Using ANCF," Multibody System Dynamics, Vol. 15, pp. 181-200. https://doi.org/10.1007/s11044-005-9004-0
  7. Yoo, W. S., Dmitrochenko, O. and Park, S. J., 2005, "A New Thin Spatial Beam Element Using the Absolute Nodal Coordinates: Application to Rotating Strip," Mechanics Based Design of Structures and Machines, Vol. 33, pp. 399-422. https://doi.org/10.1080/15367730500458267
  8. Dmitrochenko, O. and Pogorelov, D. Y., 2003, "Generalization of Plate Finite Elements for Absolute Nodal Coordinate Formulation," Multibody System Dynamics, Vol. 10, No. 1, pp. 17-43. https://doi.org/10.1023/A:1024553708730
  9. Mikkola, A. M. and Matikainen, M. K., 2006, "Development of Elastic Forces for a Large Deformation Plate Element Based on the Absolute Nodal Coordiante Formulation," Journal of Computational and Non-linear Dynamics, Vol. 1, No. 2, pp. 283-309. https://doi.org/10.1115/1.2338652
  10. Mikkola, A. M. and Shabana, A. A., 2003, "A Non-Incremental Finite Element Procedure for the Analysis of Large Deformations of Plates and Shells in Mechanical System Applications," Multibody System Dynamics, Vol. 9, pp. 283-309. https://doi.org/10.1023/A:1022950912782
  11. Kim, K. W., Lee, J. W. and Yoo, W. S., 2012, "The Motion and Deformation Rate of a Flexible hose Connected to a Mother Ship," Journal of Mechanical Science and Technology, Vol. 26, No. 3, pp. 703-710. https://doi.org/10.1007/s12206-011-1202-5
  12. Omar, M. A., Shabana, A. A., Mikkola, A. M., Loh, A. K. And Basch, R., 2004, "Multibody System Modeling of Leaf Springs," Journal of Vibration and Control, Vol. 10, pp. 1601-1638.
  13. Sugiyama, H. and Suda, Y., 2009, "Nonlinear Elastic Ring Tire Model Using the Absolute Nodal Coordinate Formulation," IMechE Journal of Multi-Body Dynamics, Vol. 223, pp. 211-219.
  14. Patel, M., Orzechowski, G., Tian, Q. and Shabana, A. A., 2016, "A New Multibody System Approach for Tire Modeling Using ANCF Finite Elements," IMechE Journal of Multi-Body Dynamics, Vol. 230, pp. 69-84.
  15. Yamashita, H. and Sugiyama, H., 2015, "High-Fidelity ANCF-LuGre Tire Model Using Continuum Mechanics Based Shear Deformable Laminated Shell Element," ECCOMAS Thematic Conference on Multibody Dynamics.
  16. Bathe, K. J., 1996, Finite Element Procedures, Englewood Cliffs, Prentice Hall.
  17. Bonet, J. and Wood, R. D., 1997, Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge: Cambridge University Press.
  18. Omar, M. A. and Shabana, A. A., 2001, "Two-Dimensional Shear Deformable Beam for Large Rotations and Deformation Problems," Journal of Sound and Vibration, Vol. 243, pp. 565-576. https://doi.org/10.1006/jsvi.2000.3416
  19. Berzeri, M. and Shabana, A. A., 2000, "Development of Simple Models for the Elastic Forces in the Absolute Nodal Coordinate Formulation," Journal of Sound and Vibration, Vol. 235, No. 4, pp. 539-565. https://doi.org/10.1006/jsvi.1999.2935
  20. Ugural, A. C. and Fenster, S. K., 1995, Advanced Strength and Applied Elasticity, Upper Saddle River, Prentice-Hall, Third edition.
  21. Kawaguti, K., Terumich, Y., Takehara, S., Kaczmarczylk, S. and Sogabe, K., 2007, "The Study of the Tether Motion with Time-Varying Length Using the Absolute Nodal Coordinate Formulation with Multiple Nonlinear Time Scales," Journal of System Design and Dynamics, Vol. 1, No. 3, pp. 491-500. https://doi.org/10.1299/jsdd.1.491
  22. Nikravesh, P. E., 1988, Computer-Aided Analysis of Mechanical Systems, Prentice-Hall Inc.
  23. Brenam, K. E., Campbell, S. L. and Petzold, L. R., 1996, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia, Pennsylvania.
  24. Newmark, N. M., 1959, "A Method of Computation for Structural Dynamics," ASCE Journal of Engineering Mechanics Division, Vol. 85, pp. 67-94.
  25. Washizu, K., 1968, Variational Methods in Elasticity and Plasticity, Pergamon Press, New York.
  26. Shabana, A. A., 2008, Computational Continuum Mechanics, Cambridge University Press, New York.
  27. Timoshenko, S. P. and Goodier, J. N., 1970, Theory of Elasticity, New York, McGraw-Hill, Inc., Third Edition.
  28. Yan, D., Liu, C., Tian, Q., Zhang, K., Liu, X. N. and Hu, G. K., 2013, "A New Curved Gradient Deficient Shell Element of Absolute Nodal Coordinate Formulation for Modeling Thin Shell Structures," Nonlinear Dynamics, Vol. 74, pp. 153-164. https://doi.org/10.1007/s11071-013-0955-z