Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Exact calculation of natural frequencies of repetitive structures

  • Williams, F.W. (Division of Structural Engineering, School of Engineering, University of Wales Cardiff) ;
  • Kennedy, D. (Division of Structural Engineering, School of Engineering, University of Wales Cardiff) ;
  • Wu, Gaofeng (Division of Structural Engineering, School of Engineering, University of Wales Cardiff) ;
  • Zhou, Jianqing (Division of Structural Engineering, School of Engineering, University of Wales Cardiff)
  • Published : 1996.09.25
⟨ Previous Next ⟩

Abstract

Finite element stiffness matrix methods are presented for finding natural frequencies (or buckling loads) and modes of repetitive structures. The usual approximate finite element formulations are included, but more relevantly they also permit the use of 'exact finite elements', which account for distributed mass exactly by solving appropriate differential equations. A transcendental eigenvalue problem results, for which all the natural frequencies are found with certainty. The calculations are performed for a single repeating portion of a rotationally or linearly (in one, two or three directions) repetitive structure. The emphasis is on rotational periodicity, for which principal advantages include: any repeating portions can be connected together, not just adjacent ones; nodes can lie on, and members along, the axis of rotational periodicity; complex arithmetic is used for brevity of presentation and speed of computation; two types of rotationally periodic substructures can be used in a multi-level manner; multi-level non-periodic substructuring is permitted within the repeating portions of parent rotationally periodic structures or substructures and; all the substructuring is exact, i.e., the same answers are obtained whether or not substructuring is used. Numerical results are given for a rotationally periodic structure by using exact finite elements and two levels of rotationally periodic substructures. The solution time is about 500 times faster than if none of the rotational periodicity had been used. The solution time would have been about ten times faster still if the software used had included all the substructuring features presented.

Keywords

References

  1. Anderson, M.S. (1981), "Buckling of periodic lattice structures", AlAA J., 19(6), 782-788.
  2. Anderson, M.S. (1982), "Vibration of prestressed periodic lattice structures", AlAA J., 20(4), 551-555.
  3. Anderson, M.S. and Williams, F.W. (1986), "Natural vibration and buckling of general periodic lattice structures", AlAA J., 24(1), 163-169.
  4. Anderson, M.S. and Williams, F.W. (1987), "BUNVIS-RG: Exact frame buckling and vibration program, with repetitive geometry and substructuring", J. Spacecraft and Rockets, 24(4), 353-361. https://doi.org/10.2514/3.25924
  5. Balasubramanian, P., Suhas, H.K. and Ramamurti, V. (1991), "Skyline solver for the static analysis of cyclic symmetric structures", Comp. and Struct., 38(3), 259-268. https://doi.org/10.1016/0045-7949(91)90104-T
  6. Capron, M.D. and Williams, F.W. (1988), "Exact dynamic stiffnesses for an axially loaded uniform Timoshenko member embedded in an elastic medium", J. Sound Vib., 124(3), 453-466. https://doi.org/10.1016/S0022-460X(88)81387-7
  7. Capron, M.D., Williams, F.W. and Symons, M.V. (1987), "A parametric study of the free vibrations of an offshore structure with piled foundations", Proc. Int. Conf. on Steel and Alum. Structs.: Steel Structs., Cardiff, 653-664. (Elsevier Applied Science, London).
  8. Henry, R. and Ferraris, G. (1983), "Substructuring and wave propagation: an efficient technique for impeller dynamic analysis", ASME Trans., Paper No. 83-GT-150, 28th Int. Gas Turbine Conf. and Exhibit, Phoenix, Arizona. 27-31 March.
  9. Howson, W.P., Banerjee, J.R. and Williams, F.W. (1983), "Concise equations and program for exact eigensolutions of plane frames including member shear", Adv. Eng. Software, 5(3), 137-141. https://doi.org/10.1016/0141-1195(83)90108-0
  10. Jemah, A.K. and Williams, F.W. (1990), "Compound stayed column for use in space", Comp. and Struct., 34(1), 171-178. https://doi.org/10.1016/0045-7949(90)90311-O
  11. Leung, A.Y.T. (1980), "Dynamic analysis of periodic structures", J. Sound Vib., 72(4), 451-467. https://doi.org/10.1016/0022-460X(80)90357-0
  12. Lunden, R. and Akesson, B.A. (1983), "Damped second-order Rayleigh-Timoshenko beam vibration in space-an exact complex dynamic member stiffness matrix", Int. J. Num. Meth. Engng, 19(3), 431-449. https://doi.org/10.1002/nme.1620190310
  13. MacNeal, R.H., Harder, R.L. and Mason, J.B. (1973), "NASTRAN cyclic symmetry capability", NASTRAN Users Experience-3rd Coll., NASA TM X-2893. 395-421.
  14. McDaniel, T.J. and Chang, K.J. (1980), "Dynamics of rotationally periodic large space structures", J. Sound Vib., 68(3), 351-368. https://doi.org/10.1016/0022-460X(80)90392-2
  15. Thomas, D.L. (1979), "Dynamics of rotationally periodic structures", Int. J. Num. Meth. Engng, 14(1), 81-102. https://doi.org/10.1002/nme.1620140107
  16. Wildheim, S.J. (1981), "Dynamics of circumferentially periodic structures", Doctoral Thesis, Linkoping Univ., Sweden.
  17. Williams, F.W. (1986a), "An algorithm for exact eigenvalue calculations for rotationally periodic structures", Int. J. Num. Meth. Engng, 23(4), 609-622. https://doi.org/10.1002/nme.1620230407
  18. Williams, F.W. (1986b), "Exact eigenvalue calculations for structures with rotationally periodic substructures", Int. J. Num. Meth. Engng, 23(4), 695-706. https://doi.org/10.1002/nme.1620230411
  19. Williams, F.W. and Wittrick, W.H. (1983), "Exact buckling and frequency calculations surveyed", J. Struct. Engng ASCE, 109(1), 169-187. https://doi.org/10.1061/(ASCE)0733-9445(1983)109:1(169)
  20. Wittrick, W.H. and Williams, F.W. (1971), "A general algorithm for computing natural frequencies of elastic structures", Q.J. Mech. Appl. Math., 24(3), 263-284. https://doi.org/10.1093/qjmam/24.3.263
  21. Wittrick, W.H. and Williams, F.W. (1973a), "An algorithm for computing critical buckling loads of elastic structures", J. Struct. Mech., 1(4), 497-518. https://doi.org/10.1080/03601217308905354
  22. Wittrick, W.H. and Williams, F.W. (1973), "New procedures for structural eigenvalue calculations", 4th Australasian Conf. on the Mechanics of Structures and Materials, U. of Queensland, Brisbane, Queensland, Austalia, 299-308.

Cited by

  1. A Reduced-Order Model of Detuned Cyclic Dynamical Systems With Geometric Modifications Using a Basis of Cyclic Modes vol.132, pp.11, 2010, https://doi.org/10.1115/1.4000805
  2. Optimal design of infinite repetitive structures vol.18, pp.2-3, 1999, https://doi.org/10.1007/BF01195995