Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Computation of dynamic stiffness and flexibility for arbitrarily shaped two-dimensional membranes

  • Chen, J.T. (Department of Harbor and River Engineering, National Taiwan Ocean University) ;
  • Chung, I.L. (Department of Harbor and River Engineering, National Taiwan Ocean University)
  • Published : 2002.04.25
⟨ Previous Next ⟩

Abstract

In this paper, dynamic stiffness and flexibility for circular membranes are analytically derived using an efficient mixed-part dual boundary element method (BEM). We employ three approaches, the complex-valued BEM, the real-part and imaginary-part BEM, to determine the dynamic stiffness and flexibility. In the analytical formulation, the continuous system for a circular membrane is transformed into a discrete system with a circulant matrix. Based on the properties of the circulant, the analytical solutions for the dynamic stiffness and flexibility are derived. In deriving the stiffness and flexibility, the spurious resonance is cancelled out. Numerical aspects are discussed and emphasized. The problem of numerical instability due to division by zero is avoided by choosing additional constraints from the information of real and imaginary parts in the dual formulation. For the overdetermined system, the least squares method is considered to determine the dynamic stiffness and flexibility. A general purpose program has been developed to test several examples including circular and square cases.

Keywords

Acknowledgement

Supported by : National Science Council

References

  1. Banerjee, J.R. (1996), "Exact dynamic stiffness matrix for composite Timoshenko beams with applications", J. Sound and Vibration, 194, 573-585. https://doi.org/10.1006/jsvi.1996.0378
  2. Banerjee, J.R. (1998), "Free vibration of axially loaded composite Timoshenko beams using the dynamicstiffness matrix method", Comp. and Struct., 69, 197-208. https://doi.org/10.1016/S0045-7949(98)00114-X
  3. Chang, C.M. (1998), "Analysis of natural frequencies and natural modes for rod and beam problems usingMultiple Reciprocity Method (MRM)", Master Thesis, Department of Harbor and River Engineering, NationalTaiwan Ocean University, Taiwan.
  4. Chang, J.R., Yeih, W. and Chen, J.T. (1999), "Determination of natural frequencies and natural modes using thedual BEM in conjunction with the domain partition technique", Comp. Mech., 24(1), 29-40. https://doi.org/10.1007/s004660050435
  5. Chen, J.T. and Chen, K.H. (1998), "Dual integral formulation for determining the acoustic modes of a twodimensionalcavity with a degenerate boundary", Engineering Analysis with Boundary Elements, 21(2), 105-116. https://doi.org/10.1016/S0955-7997(97)00094-5
  6. Chen, J.T., Huang, C.X. and Chen, K.H. (1999), "Determination of spurious eigenvalues and multiplicities oftrue eigenvalues using the real-part dual BEM", Comp. Mech., 24(1), 41-51. https://doi.org/10.1007/s004660050436
  7. Chen, J.T., Kuo, S.R. and Huang, C.X. (2001), On the True and Spurious Eigensolutions Using Circulants forReal-Part BEM, IUTAM/IACM/IABEM Symposium on BEM, Cracow, Poland, Kluwer Press, 75-85.
  8. Chen, J.T., Huang, C.X. and Wong, F.C. (2000), "Determination of spurious eigenvalues and multiplicities oftrue eigenvalues in the dual multiple reciprocity method using the singular value decomposition technique", J.Sound Vibration, 230(2), 203-219. https://doi.org/10.1006/jsvi.1999.2342
  9. Chen, J.T. and Wong, F.C. (1998), "Dual formulation of multiple reciprocity method for the acoustic mode of acavity with a thin partition", J. Sound and Vibration, 217(1), 75-95. https://doi.org/10.1006/jsvi.1998.1743
  10. Chiba, M. (1997), "The influence of elastic bottom plate motion on the resonant response of a liquid free surfacein a cylindrical container : a linear analysis", J. Sound and Vibration, 202(4), 417-426. https://doi.org/10.1006/jsvi.1996.0855
  11. Chung, I.L. (2000), "Derivation of dynamic stiffness and flexibility using dual BEM", Master Thesis,Department of Harbor and River Engineering, National Taiwan Ocean University, Taiwan.
  12. Clough, R.W. and Penzien, J. (1975), Dynamics of Structures, McGraw-Hill, New York.
  13. Hibbeler, R.C. (1997), Structural Analysis, Prentice Hall, New York.
  14. Ishibashi, Y. and Orihara, H. (1996), "Nonlinear dielectric responses in resonant systems", Physica B, 219, 626-628. https://doi.org/10.1016/0921-4526(95)00833-0
  15. Kang, S.W., Lee, J.M. and Kang, Y.J. (1999), "Vibration analysis of arbitrarily shaped membranes using nondimensionaldynamic influence function", J. Sound and Vibration, 221(1), 117-132. https://doi.org/10.1006/jsvi.1998.2009
  16. Liou, D.Y., Chen, J.T. and Chen, K.H. (1999), "A new method for determining the acoustic modes of a twodimensionalsound field", J. the Chinese Institute of Civil and Hydraulic Engineering, 11(2), 299-310. (in Chinese)
  17. Luco, J.E. and Westmann, R.A. (1972), "Dynamic response of a rigid footing bounded to an elastic half space",J. Appl. Mech., 39, 527-534, 1972. https://doi.org/10.1115/1.3422711
  18. Lysmer, J. and Kuhlemeyer, R.L. (1969), "Finite dynamic model for infinite media", J. Eng. Mech. Div., 95,859-877.
  19. Paz, M. (1973), "Mathematical observations in structural dynamics", Comp. Struct., 3, 385-396. https://doi.org/10.1016/0045-7949(73)90025-4
  20. Paz, M. and Lam. D. (1975), "Power series expansion of the general stiffness matrix for beam elements", Int. J.Num. Meth. Engng., 9, 449-459. https://doi.org/10.1002/nme.1620090212
  21. Reid, C. and Whineray, S. (1995), "The resonant response of a simple harmonic half-oscillator", Physics LettersA, 199, 49-54. https://doi.org/10.1016/0375-9601(95)00038-5
  22. Samaranayake, S., Samaranayake, G. and Bajaj, A.K. (2000), "Resonant vibrations in harmonically excitedweakly coupled mechanical systems with cyclic symmetry", Chaos Solitons and Fractals, 11, 1519-1534. https://doi.org/10.1016/S0960-0779(99)00075-2
  23. Wearing, J.L. and Bettahar, O. (1996), "The analysis of plate bending problems using the regular direct boundaryelement method", Engineering Analysis with Boundary Elements, 16, 261-271. https://doi.org/10.1016/0955-7997(96)86002-4
  24. Wolf, J.P. and Song, C. (1994), "Dynamic-stiffness matrix in time domain of unbounded medium byinfinitesimal finite element", Earthq. Eng. and Struct. Dyn., 23, 1181-1198. https://doi.org/10.1002/eqe.4290231103
  25. Wu, Y.C. (1999), "Applications of the generalized singular value decompsition method to the eigenproblem of the Helmholtz equation", Master Thesis, Department of Harbor and River Engineering, National TaiwanOcean University, Taiwan.
  26. Yeih, W., Chang, J.R., Chang, C.M. and Chen, J.T. (1999), "Applications of dual MRM for determining thenatural frequencies and natural modes of a rod using the singular value decomposition method", Advances inEngineering Software, 30(7), 459-468. https://doi.org/10.1016/S0965-9978(98)00130-6
  27. Yeih, W., Chen, J.T. and Chang, C.M. (1999), "Applications of dual MRM for determining the naturalfrequencies and natural modes of an Euler-Bernoulli beam using the singular value decomposition method",Engineering Analysis with Boundary Elements, 23(4), 339-360. https://doi.org/10.1016/S0955-7997(98)00084-8
  28. Zhao, J.X., Carr, A.J. and Moss, P.J. (1997), "Calculating the dynamic stiffness matrix of 2-D foundations bydiscrete wave number indirect boundary element methods", Earthq. Eng. and Struct. Dyn., 26, 115-133. https://doi.org/10.1002/(SICI)1096-9845(199701)26:1<115::AID-EQE626>3.0.CO;2-0