The member stiffness determinant and its uses for the transcendental eigenproblems of structural engineering and other disciplines

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 459, Issue 2032, 2003, Pages 1001-1019

  Williams, Frederic W ^a   Kennedy, David ^b   Djoudi, Mohamed S ^b  

^a CITY UNIVERSITY OF HONG KONG   (Hong Kong)

^b CARDIFF UNIVERSITY   (United Kingdom)

Author keywords

Buckling; Eigenproblem; Structures; Transcendental; Vibration; Wittrick Williams algorithm

Indexed keywords

ACCURACY; ALGORITHM; ARTICLE; ENGINEERING; FINITE ELEMENT ANALYSIS; MATHEMATICAL ANALYSIS; MATHEMATICAL COMPUTING; MATHEMATICAL MODEL; RIGIDITY; STRUCTURE ANALYSIS;

EID: 1542616824     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2002.1074     Document Type: Article

References (42)

1. Åkesson, B. Å. 1976 PFVIBAT - a computer program for plane frame vibration analysis by an exact method. Int. J. Numer. Meth. Engng 10, 1221-1231.
2. Anderson, M. S., Williams, F. W. & Wright, C. J. 1983 Buckling and vibration of any prismatic assembly of shear and compression loaded anisotropic plates with an arbitrary supporting structure. Int. J. Mech. Sci. 25, 585-596.
3. Balakrishnan, A. V. 1993 On the Wittrick-Williams algorithm for modes of flexible structures. Optim. Meth. Software 2, 177-187.
4. Balakrishnan, A. V. 1995 Generalization of the Wittrick-Williams formula for counting modes of flexible structures. J. Guid. Control Dynam. 18, 1410-1415.
5. Butler, R. & Williams, F. W. 1992 Optimum design using VICONOPT, a buckling and strength constraint program for prismatic assemblies of anisotropic plates. Comput. Struct. 43, 699-708.
6. Clough, R. W. & Penzien, J. 1993 Dynamics of structures, pp. 184-189. McGraw-Hill.
7. Edwards, D. A., Williams, F. W. & Kennedy, D. 1998 Cost optimization of stiffened panels using VICONOPT. AIAA J. 36, 267-272.
8. Frid, A. 1989 Fluid vibration in piping systems: a structural mechanics approach. I. Theory. J. Sound Vib. 133, 423-438.
9. Frid, A. 1990 Stationary and transient fluid vibration in piping system using the structural mechanics approach. Publ. PVP, vol. 202, pp. 1-6. New York: ASME.
10. Gupta, A. K. & Howson, W. P. 1994 Exact natural frequencies of plane structures composed of slender elastic curved members. J. Sound Vib. 175, 145-147.
11. Hopper, C. T. & Williams, F. W. 1977 Mode finding in non-linear structural eigenvalue calculations. J. Struct. Mech. 5, 255-278.
12. Howson, W. P. 1979 A compact method for computing the eigenvalues and eigenvectors of plane frames. Adv. Engng Software 1, 181-190.
13. Howson, W. P. & Jemah, A. K. 1999a Exact dynamic stiffness method for planar natural frequencies of curved Timoshenko beams. J. Mech. Engng Sci. 213, 687-696.
14. Howson, W. P. & Jemah, A. K. 1999b Exact out-of-plane natural frequencies of curved Timoshenko beams. J. Engng Mech. 125, 19-25.
15. Howson, W. P. & Williams, F. W. 1973 Natural frequencies of frames with axially loaded Timoshenko members. J. Sound Vib. 26, 503-515.
16. Howson, W. P., Banerjee, J. R. & Williams, F. W. 1983 Concise equations and program for exact eigensolutions of plane frames including member shear. Adv. Engng Software 5, 137-141.
17. Howson, W. P., Jemah, A. K. & Zhou, J. Q. 1995 Exact natural frequencies for out-of-plane motion of plane structures composed of curved beam members. Comput. Struct. 55, 989-995.
18. Kennedy, D. & Williams, F. W. 1991 More efficient use of determinants to solve transcendental structural eigenvalue problems reliably. Comput. Struct. 41, 973-979.
19. Kennedy, D., Williams, F. W. & Anderson, M. S. 1994 Buckling and vibration analysis of laminated panels using VICONOPT. J. Aerospace Engng 7, 245-262.
20. Lightfoot, E. 1980 Exact straight-line elements. J. Strain Analysis 15, 89-96.
21. Mikhailov, M. D. & Ozisik, M. N. 1984 Unified analysis and solutions of heat and mass diffusion. Wiley.
22. Mikhailov, M. D. & Vulchanov, N. L. 1983 Computational procedure for Sturm-Liouville problems. J. Computat. Phys. 50, 323-336.
23. Mikhailov, M. D., Ozisik, M. N. & Vulchanov, N. L. 1983 Diffusion in composite layers with automatic solution of the eigenvalue problem. Int. J. Heat Mass Transfer 26, 1131-1141.
24. Powell, S. M., Kennedy, D. & Williams, F. W. 1997 Efficient multi-level substructuring with constraints for buckling and vibration analysis of prismatic plate assemblies. Int. J. Mech. Sci. 39, 795-805.
25. Ronagh, H. R., Lawther, R. & Williams, F. W. 1995 Calculation of eigenvectors with uniform accuracy. J. Engng Mech. 121, 948-955.
26. Simpson, A. & Tabarrok, B. 1968 On Kron's eigenvalue procedure and related methods of frequency analysis. Q. J. Mech. Appl. Math. 21, 1-39.
27. Smith, H. A., Sorensen, D. C. & Singh, R. K. 1993 A Lanczos-based technique for exact vibration analysis of skeletal structures. Int. J. Numer. Meth. Engng 36, 187-200.
28. Williams, F. W. 1971 Natural frequencies of repetitive structures. Q. J. Mech. Appl. Math. 24, 285-310.
29. Williams, F. W. 1973 Comparison between sparse stiffness matrix and sub-structure methods. Int. J. Numer. Meth. Engng 5, 383-394.
30. Williams, F. W. 1986 An algorithm for exact eigenvalue calculations for rotationally periodic structures. Int. J. Numer. Meth. Engng 23, 609-622.
31. Williams, F. W. & Kennedy, D. 1988 Reliable use of determinants to solve non-linear structural eigenvalue problems efficiently. Int. J. Numer. Meth. Engng 26, 1825-1841.
32. Williams, F. W. & Wittrick, W. H. 1970 An automatic computational procedure for calculating natural frequencies of skeletal structures. Int. J. Mech. Sci. 12, 781-791.
33. Williams, F. W. & Wittrick, W. H. 1983 Exact buckling and frequency calculations surveyed. J. Struct. Engng 109, 169-187.
34. Williams, F. W., Ouyang, H. J., Kennedy, D. & York, C. B. 1995 Wave propagation along longitudinally periodically supported or stiffened prismatic plate assemblies. J. Sound Vib. 186, 197-205.
35. Wittrick, W. H. & Williams, F. W. 1971 A general algorithm for computing natural frequencies of elastic structures. Q. J. Mech. Appl. Math. 24, 263-284.
36. Wittrick, W. H. & Williams, F. W. 1973a An algorithm for computing critical buckling loads of elastic structures. J. Struct. Mech. 1, 497-518.
37. Wittrick, W. H. & Williams, F. W. 1973b New procedures for structural eigenvalue calculations. In Proc. 4th Australasian Conf. on the Mechanics of Structures and Materials, Brisbane, Australia, August 1973, pp. 299-308.
38. Wittrick, W. H. & Williams, F. W. 1974 Buckling and vibration of anisotropic or isotropic plate assemblies under combined loadings. Int. J. Mech. Sci. 16, 209-239.
39. Wittrick, W. H. & Williams, F. W. 1982 On the free vibration analysis of spinning structures by using discrete or distributed mass models. J. Sound Vib. 82, 1-15.
40. Ye, J. & Williams, F. W. 1995 A successive bounding method to find the exact eigenvalues of transcendental stiffness matrix formulations. Int. J. Numer. Meth. Engng 38, 1057-1067.
41. ∞ filtering with secure eigenvalue calculation and precise integration. Int. J. Numer. Meth. Engng 146, 1017-1030.
42. Zhong, W., Williams, F. W. & Bennett, P. N. 1997 Extension of the Wittrick-Williams algorithm to mixed variable systems. J. Vib. Acoust. 119, 334-340.