DOI QR코드

DOI QR Code

Radial basis collocation method for dynamic analysis of axially moving beams

  • Wang, Lihua (School of Aerospace Engineering and Applied Mechanics, Tongji University) ;
  • Chen, Jiun-Shyan (Civil & Environmental Engineering Department, University of California Los Angeles (UCLA)) ;
  • Hu, Hsin-Yun (Mathematics Department, Tunghai University)
  • 투고 : 2009.09.01
  • 심사 : 2009.09.17
  • 발행 : 2009.12.25

초록

We introduce a radial basis collocation method to solve axially moving beam problems which involve $2^{nd}$ order differentiation in time and $4^{th}$ order differentiation in space. The discrete equation is constructed based on the strong form of the governing equation. The employment of multiquadrics radial basis function allows approximation of higher order derivatives in the strong form. Unlike the other approximation functions used in the meshfree methods, such as the moving least-squares approximation, $4^{th}$ order derivative of multiquadrics radial basis function is straightforward. We also show that the standard weighted boundary collocation approach for imposition of boundary conditions in static problems yields significant errors in the transient problems. This inaccuracy in dynamic problems can be corrected by a statically condensed semi-discrete equation resulting from an exact imposition of boundary conditions. The effectiveness of this approach is examined in the numerical examples.

키워드

참고문헌

  1. Chen, J.S., Hu, W. and Hu, H.Y. (2008), "Reproducing Kernel Enhanced Local Radial Basis Collocation Method", Int. J. Numer. Method. E., 75, 600-627. https://doi.org/10.1002/nme.2269
  2. Chen, J.S., Wang, L., Hu, H.Y. and Chi, S.W. (2009), "Subdomain Radial Basis Collocation Method for Heterogeneous Media", J. Numer. Method. E., 80, 163-190. https://doi.org/10.1002/nme.2624
  3. Clough, W.R. and Penzien, J. (2003), Dynamics of Structures, 3rd Edition, Computers and Structures, Inc.
  4. Fasshauer, G.E. (1999), "Solving differential equations with radial basis functions: multilevel methods and smoothing", Mech. Adv. Mater. Struc., 11(2/3), 139-159.
  5. Ferreira, A.J.M. (2003), "Thick composite beam analysis using a global meshless approximation based on radial basis functions", Mech. Adv. Mater. Struc., 10, 271-284. https://doi.org/10.1080/15376490306743
  6. Gurgoze, M. (1999), "Transverse vibration of a flexible beam sliding through a prismatic joint", J. Sound Vib., 223(3), 467-482. https://doi.org/10.1006/jsvi.1999.2155
  7. Hardy, R.L. (1971), "Multiquadric equations of topography and other irregular surfaces", J. Geophys. Res., 176, 1905-1915.
  8. Hon, Y.C. and Schaback, R. (2001), "On unsymmetric collocation by radial basis functions", Appl. Math. Comput., 119(2-3), 177-186. https://doi.org/10.1016/S0096-3003(99)00255-6
  9. Hu, H.Y., Chen, J.S. and Hu, W. (2007), "Weighted radial basis collocation method for boundary value problems", Int. J. Numer. Method. E., 69, 2736-2757. https://doi.org/10.1002/nme.1877
  10. Kansa, E.J. (1990a), "Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics - i. surface approximations and partial derivative estimates", Comput. Math. Appl., 19, 127-145.
  11. Kansa, E.J. (1990b), "Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics - ii. solutions to parabolic, hyperbolic and elliptic partial differential equations", Comput. Math. Appl., 19, 147-161.
  12. Kong, L. and Parker, R.G. (2004), "Approximate eigensolutions of axially moving beams with small flexural stiffness", J. Sound Vib., 276, 459-469. https://doi.org/10.1016/j.jsv.2003.11.027
  13. Lee, U. and Jang, I. (2007), "On the boundary for axially moving beams", J. Sound Vib., 306, 675-690. https://doi.org/10.1016/j.jsv.2007.06.039
  14. Lee, U. and Oh, H. (2005), "Dynamics of an axially moving viscoelastic beam subject to axial tension", Int. J. Solids Struct., 42, 2381-2398. https://doi.org/10.1016/j.ijsolstr.2004.09.026
  15. Liu, Y., Liew, K.M., Hon, Y.C. and Zhang, X. (2005), "Numerical simulation and analysis of an electroactuated beam using a radial basis function", Smart Mater. Struct., 14, 1163-1171. https://doi.org/10.1088/0964-1726/14/6/009
  16. Madych, W.R. and Nelson, S.A. (1992), "Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation", J. Approx. Theory, 70, 94-114. https://doi.org/10.1016/0021-9045(92)90058-V
  17. Ozkaya, E. and Pakdemirli, M. (2002), "Group-theoretic approach to axially accelerating beam problem", Acta Mech., 155, 111-123. https://doi.org/10.1007/BF01170843
  18. Raju, I.S., Phillips, D.R. and Krishnamurthy, T. (2003), "Meshless local Petrov-Galerkin Euler-Bernoulli beam problems: a radial basis function approach", Proceedings of the 44th AIAA/ASME/AHS Structure, Structure Dynamics, and Materials Conference, Norfolk, April.
  19. Rao, G.V. (1992), "Linear dynamics and active control of an elastically supported traveling string", Comput. Struct., 43(6), 1041-1049. https://doi.org/10.1016/0045-7949(92)90004-J
  20. Stylianou, M. and Tabarrok, B. (1994), "Finite element analysis of an axially moving beam, part I: time integration", J. Sound Vib., 178(4), 433-453. https://doi.org/10.1006/jsvi.1994.1497
  21. Sylla, M. and Asseke, B. (2008), "Dynamics of a rotating flexible and symmetric spacecraft using impedance matrix in terms of the flexible appendages cantilever modes", Multibody Syst. Dyn., 19, 345-364. https://doi.org/10.1007/s11044-007-9102-2
  22. Tadikonda, S.S.K. and Baruh, H. (1992), "Dynamics and control of a translating flexible beam with a prismatic joint", J. Dyn. Syst-T. ASME, 114(3), 422-427. https://doi.org/10.1115/1.2897364
  23. Tiago, C.M. and Leitao, V.M.A. (2006), "Application of radial basis functions to linear and nonlinear structural analysis problems", Comput. Math. Appl., 51, 1311-1334. https://doi.org/10.1016/j.camwa.2006.04.008
  24. Wang, L.H., Hu, Z.D., Zhong, Z. and Ju, J.W. (2009), "Hamiltonian dynamic analysis of an axially translating beam featuring time-variant velocity", Acta Mech., 206(3-4), 149-161. https://doi.org/10.1007/s00707-008-0104-9
  25. Wickert, J.A. and Mote, C.D., Jr. (1991), "Response and discretization methods for axially moving materials", Appl. Mech. Rev., 44(11S), 279-284. https://doi.org/10.1115/1.3121365
  26. Wu, L.Y., Chung, L.L. and Huang, H.H. (2008), "Radial spline collocation method for analysis of beams", Appl. Math. Comput., 201, 184-199. https://doi.org/10.1016/j.amc.2007.12.012
  27. Yoo, H.H. (1995), "Dynamics of flexible beams undergoing overall motions", J. Sound Vib., 181(2), 261-278. https://doi.org/10.1006/jsvi.1995.0139
  28. Yuh, J. and Young, T. (1990), "Dynamic modeling of an axially moving beam in rotation: simulation and experiment", J. Dyn. Syst-T. ASME, 113, 34-40.
  29. Zhu, W.D. and Ni, J. (2000), "Energetics and stability of translating media with an arbitrarily varying length", J. Vib. Acoust., 122, 295-304. https://doi.org/10.1115/1.1303003

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