Advertisement

Sensitivity Analysis for Dynamic Stability Problems

  • Pauli Pedersen
Part of the International Centre for Mechanical Sciences book series (CISM, volume 436)

Abstract

These notes are neither chapters in a textbook, nor short research papers. They are something in between and they include results known for some years as well as just published results. The six notes are written to be fairly independent without extensive mutual reference and the page limit for each note is set to 10. This page limit means that extensive examples are not included.

Keywords

Stability Diagram Follower Force Double Eigenvalue Characteristic Multiplier Mutual Energy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arscott, F. M. (1964). Periodic differential equations. Pergamon. 281 pages.Google Scholar
  2. Benjamin, T. B. (1961). Dynamics of a system of articulated pipes conveying fluid. Proc. Roy. Soc. Lond. A 261: 457–499.CrossRefMATHMathSciNetGoogle Scholar
  3. Bishop, R. E. D., and Fawzy, I. (1976). Free and forced oscillation of a vertical tube containing a flowing fluid. Phil. Trans. Roy. Soc. Lond. A 284: 1–47.CrossRefGoogle Scholar
  4. Bolotin, V. V., and Zhinzher, N. I. (1969). Effects of damping on stability of elastic systems subjected to non-conservative forces. Int. J. Solids and Structures 965–989.Google Scholar
  5. Bolotin, V. V. (1963). Nonconservative Problems of the Theory of Elastic Stability. Pergamon. 324 pages.Google Scholar
  6. Bolotin, V. V. (1964). The Dynamic Stability of Elastic Systems. Holden-Day. 451 pages.Google Scholar
  7. Bunkov, V. G. (1969). Calculation of optimal flutter characteristics by gradient method (in russian). TSAGI 1166.Google Scholar
  8. Cartmell, M. P. (1990). Introduction to Linear, Parametric and Nonlinear Vibrations. London: Chapman and Hall.MATHGoogle Scholar
  9. Cesari, L. (1964). Asymptotic behaviour and stability problems in ordinary differential equations. Academic Press.Google Scholar
  10. Epstein, B., and Barakat, R. (1977). Perturbation solutions of the Carson-Cambi equation. J. Franklin Institute 303: 177–188.CrossRefMATHGoogle Scholar
  11. Floquet, G. (1883). Sur les equation differentielles lineaires coefficients periodiques. Annales de Ecole Normal Superior 2 (12): 47–89.MathSciNetGoogle Scholar
  12. Frederiksen, P. S. (1997). Experimental procedure and results for the identification of elastic constants of thick orthotropic plates. J. Composite Materials 31: 360–382.CrossRefGoogle Scholar
  13. Fu, F. C. L., and Nemat-Nasser, S. (1972). Stability of solution of systems of linear differential equations with harmonic coefficients. AIAAJ. 10 (1): 30–36.CrossRefMATHMathSciNetGoogle Scholar
  14. Gregory, R. W., and Paidoussis, M. P. (1966). Unstable oscillation of tubular cantilevers conveying fluid. ii. experiments. Proc. of the Royal Society Lond. A 293: 528–542.CrossRefGoogle Scholar
  15. Grimshaw, R. (1990). Nonlinear ordinary differential equations. Blackwell.Google Scholar
  16. Hansen, J. M. (1985). Stability diagrams for coupled mathieu-equations. Ingenieur-Archiv 55: 463–473.CrossRefMATHGoogle Scholar
  17. Herrmann, G., and Jong, I. C. (1965). On the destabilizing effect of damping in nonconservative elastic systems. J Appl. Mech. 32: 592–597.CrossRefMathSciNetGoogle Scholar
  18. Hermann, G. (1967). Stability of equilibrium of elastic systems subjected to nonconservative forces. Appl. Mech. Revs 20: 103–108.MathSciNetGoogle Scholar
  19. Hill, G. W. (1886). On the part of the motion of the lunar perigee. Acta Math. 8: 1–36.CrossRefMATHMathSciNetGoogle Scholar
  20. Hsu, C. S. (1961). On a restricted class of coupled Hill’s equations and some applications. J. of Appl. Mech. 28: 551–556.CrossRefMATHGoogle Scholar
  21. Huseyin, K. (1978). Vibrations and stability of multiple parameter systems. Sijthoff and Nordhoff.Google Scholar
  22. Jones, R. M. (1975). Mechanics of Composite Materials. McGraw-Hill.Google Scholar
  23. Jorgensen, O. (1991). Optimization of the flutter load by material orientation. Mechanics of Structures and Machines 19 (3): 411–436.CrossRefGoogle Scholar
  24. Langthjem, M. A., and Sugiyama, Y. (1999a). Optimal design of Beck’s column with a constraint on the static buckling load. Structural Optimization 18: 228–235.CrossRefGoogle Scholar
  25. Langthjem, M. A., and Sugiyama, Y. (1999b). Optimal shape design against flutter of a cantilevered column with an end-mass of finite size subjected to a non-conservative load. J. of Sound and Vibration 226 (1): 1–23.CrossRefGoogle Scholar
  26. Langthjem, M. A., and Sugiyama, Y. (2000a). Dynamic stability of columns subjected to follower loads: A survey. J. of Sound and Vibration 238 (5): 809–851.CrossRefGoogle Scholar
  27. Langthjem, M. A., and Sugiyama, Y. (2000b). Optimal design of cantilevered columns under the combined action of conservative and nonconservative loads. part i the undamped case. Computers and Structures 74: 385–398.CrossRefGoogle Scholar
  28. Langthjem, M. A., and Sugiyama, Y. (2000c). Optimal design of cantilevered columns under the combined action of conservative and nonconservative loads. part ii the damped case. Computers and Structures 74: 399–408.CrossRefGoogle Scholar
  29. Langthjem, M. A. (1996). Dynamics, Stability and Optimal Design of Structures with Fluid Interaction. DCAMM Sxx, Solid Mechanics, DTU - thesis for the Ph.D.Google Scholar
  30. Liapunov, A. (1949). Probleme general de la stabilite du mouvement, volume 17 of Ann. Math. Studies. Princeton Univ. Press.Google Scholar
  31. Lindh, K. G., and Likins, P. W. (1970). Infinite determinant methods for stability analysis of periodic-coefficient differential equations. AIAA J. 8 (4): 680–686.CrossRefMATHGoogle Scholar
  32. Lottati, I., and Kornecki, A. (1986). The effects of an elastic foundation and of dissipative forces on the stability of fluid-conveying pipes. J. of Sound and libration 109: 327–338.CrossRefGoogle Scholar
  33. Magnus, W., and Winkler, S. (1966). Hill’s equation. 125 pages.Google Scholar
  34. Markworth, N. J., and Petersen, C. (1987). Identifikation of Materialeparametre for Fiberbaserede Laminatplader (McS thesis in Danish). Solid Mechanics, DTU.Google Scholar
  35. Mathieu, E. (1868). Memoire sur le mouvement vibratoire d’une membrane de forme elliptique. J. de Math. Pures et Appliquees (J. de Liouville) 13: 137–203.MATHGoogle Scholar
  36. McLachlan, N. W. (1947). Theory and application of Mathieu functions. 401 pages.Google Scholar
  37. Meixner, J., and Schaefke, F. W. (1954). Mathieusche funktionen und sphaeroidfunktionen. 414 pages.Google Scholar
  38. Mettler, E. (1959). Stabilitaetsfragen bei freien schwingungen mechanicher systeme. Ingenieur-Archiv 28: 213–228.CrossRefMATHMathSciNetGoogle Scholar
  39. Nayfeh, A. H., and Mook, D. T. (1979). Nonlinear oscillations. Wiley.Google Scholar
  40. Niordson, F. I. (1965). On the optimal design of a vibrating beam. Quarterly of Applied Mathematics 23: 47–53.MathSciNetGoogle Scholar
  41. Olhoff, N. (1985). Optimal design with respect to structural eigenvalues. In Rimrott, F. P. J., and Tabarrot, B., eds., IUTAM Congress, 133–149. Canada: North-Holland.Google Scholar
  42. Otterbein, S. (1982). Stabilisierung des n-pendels und der indische seiltrick. Arch. for Rational Mechanics and Analysis 78: 381–393.CrossRefMATHMathSciNetGoogle Scholar
  43. Paidoussis, M. P., and Li, G. X. (1993). Pipes conveying fluid: a model dynamical problem. J. of Fluids and Structures 7: 137–204.CrossRefGoogle Scholar
  44. Panovko, Y. G., and Gubanova, I. I. (1964). Stability and Oscillations of Elastic Systems. Consultants Bureau. 290 pages.Google Scholar
  45. Pedersen, P., and Seyranian, A. P. (1983). Sensitivity analysis for problems of dynamic stability. Int. J. of Solids and Structures 19 (4): 315–335.CrossRefMATHGoogle Scholar
  46. Pedersen, P. (1980). A quantitative stability analysis of the solutions to the Carson-Cambi equation. J. of the Franklin Institute 309 (5): 359–367.CrossRefMATHGoogle Scholar
  47. Pedersen, P. (1981). The integrated approach of fern-slp for solving problems of optimal design. In Haug, E., and Cea, J., eds., Optimization Distributed Parameter Structures, volume 49 of NATO ASI series, 757–780. Sijthoff and Noordhoff.Google Scholar
  48. Pedersen, P. (1982–83). Design with several eigenvalue constraints by finite elements and linear programming. J. Structural Mechanics 10(3):243–271.Google Scholar
  49. Pedersen, P. (1983). A unified approach to optimal design. In Eschenauer, H., and Olhoff, N., eds., Optimization Methods in Structural Design, 182–187. Siegen, Germany: B.I. Wissenschafsverlag.Google Scholar
  50. Pedersen, P. (1984). Sensitivity analysis for non-selfadjoint problems. In Komkov, V., ed., Sensitivity of Functionals with Applications to Engineering Sciences, volume 1086 of Lecture Notes in Mathematics, 119–130. New York, USA: American Mathematical Society.Google Scholar
  51. Pedersen, P. (1985). On stability diagrams for damped Hill equations. Quarterly of Applied Mathematics 42 (4): 477–495.MATHMathSciNetGoogle Scholar
  52. Pedersen, P. (1987). On sensitivity analysis of optimal design of specially orthotropic laminates. Engineering Optimization 11: 305–316.CrossRefGoogle Scholar
  53. Pedersen, P. (1999). Sensitivity analysis and inverse problems for laminates and materials. In Mota Soares, C. A., Mota Soares, C. M., and Freitas, M. J. M., eds., Mechanics of Composite Materials and Structures, NATO ASI series, 453–463. Troia, Portugal: Kluwer.Google Scholar
  54. Rayleigh, L. (1887). Maintenance of vibrations by forces of double frequency, and propagation of waves through a medium with a periodic structure. P.M. 24: 145.MATHGoogle Scholar
  55. Seyranian, A. P., and Pedersen, P. (1995). On two effects in fluid/structure interaction theory. In Bearman, P. W., ed., 6th Int. Conference on Fluid-Induced Vibration, 565–576. London, UK: Balkema.Google Scholar
  56. Seyranian, A. P., Lund, E., and Olhoff, N. (1994). Multiple eigenvalues in structural optimization problems. Structural Optimization 8 (4): 207–227.CrossRefGoogle Scholar
  57. Seyranian, A. P., Solem, F., and Pedersen, P. (1999). Stability analysis for multi-parameter periodic systems. Archive of Applied Mechanics 69: 160–180.CrossRefMATHGoogle Scholar
  58. Seyranian, A. P., Solem, F., and Pedersen, P. (2000). Multi-parameter linear periodic systems: Sensitivity analysis and applications. J. of Sound and Vibration 229 (1): 89–111.CrossRefMATHMathSciNetGoogle Scholar
  59. Seyranian, A. P. (1990). Destabilization paradox in stability problems of nonconservative systems. Advances in Mechanics 13 (2): 89–124.MathSciNetGoogle Scholar
  60. Seyranian, A. P. (1991). Stability and catastrophes of vibrating systems depending on parameters - Lecture notes. DCAMM. 76 pages.Google Scholar
  61. Seyranian, A. P. (1993). Sensitivity analysis of multiple eigenvalues. Mechanics of Structures and Machines 21 (2): 261–284.CrossRefMathSciNetGoogle Scholar
  62. Sinha, S. C., and Wu, D. H. (1991). An efficient computational scheme for the analysis of periodic systems. J. of Sound and Vibration 151 (1): 91–117.CrossRefMathSciNetGoogle Scholar
  63. Smith, G. (1975). Parameterregte schwingungen. VEB Berlin. 313 pages.Google Scholar
  64. Stoker, J. J. (1950). Nonlinear Vibrations in Mechanical and Electrical Systems. Interscience. 273 pages.Google Scholar
  65. Strutt, M. J. O. (1932). Lamesche-Mathieusche-und verwandte funktionen in physik und technik. Springer. 116 pages.Google Scholar
  66. Sugiyama, Y, and Noda, T. (1981). Studies on stability of two-degrees-of-freedom articulated pipes conveying fluid. Bulletin of JSME 24 (194–9): 1354–1362.CrossRefGoogle Scholar
  67. Thomsen, J. J. (1997). Vibrations and Stability, Order and Chaos. McGraw-Hill. 323 pages.Google Scholar
  68. Vishik, M. I., and Lyusternik, L. A. (1960). Solution of some perturbation problems in the case of matrices and self-adjoint or non-self-adjoint equations. Russian Math. Surveys 15 (3): 1–73.CrossRefMATHMathSciNetGoogle Scholar
  69. Wittrick, W. H. (1962). Rates of change of eigenvalues, with reference to buckling and vibration problems. J. Royal Aeronautical Soc. 66: 590–591.Google Scholar
  70. Wolfram, S. (1991). Mathematica: a system for doing mathematics by computer. Addison-Wesley.Google Scholar
  71. Yakubovitch, V. A., and Starzhinskii, V. M. (1975). Linear differential equations with periodic coefficients,volume 1. Wiley.Google Scholar
  72. Yakubovitch, V. A., and Starzhinskii, V. M. (1987). Parametric resonance in linear systems (in Russian). Nauka.Google Scholar
  73. Ziegler, H. (1952). Die stabilitaetskriterien der elastomechanik. Ing.-Arch. 20: 49–56.CrossRefMATHMathSciNetGoogle Scholar
  74. Ziegler, H. (1968). Principles of Structural Stability. Blaisdell. 150 pages.Google Scholar

Copyright information

© Springer-Verlag Wien 2002

Authors and Affiliations

  • Pauli Pedersen
    • 1
  1. 1.Department of Mechanical Engineering, Solid MechanicsTechnical University of DenmarkKgs.LyngbyDenmark

Personalised recommendations