Dynamic Stability of Structures

  • V. V. Bolotin
Part of the International Centre for Mechanical Sciences book series (CISM, volume 342)


A survey of the theory of dynamic stability of structures and structural components is presented including both the classical analysis within the framework of nonlinear mechanics and the numerical simulation with the discussion of computational results in terms of the up-to-date approaches to the theory of dynamic systems. Typical problems of dynamic stability of nonlinear structures are considered: parametric resonances in structures and structural components under time periodic excitation; instabilities of structures under essentially non-conservative forces; various instabilities in structure-flow interacting systems (wing flutter, panel flutter, etc.); rotor whirling due to internal damping and related factors, etc. The introductory part of lecturer’s notes concludes with a survey of methods to analyse stability and post-critical behaviour of mechanical (including — continuous) systems. Special attention is paid to the interaction between forced and parametrically excited vibrations in continuous systems; to the postcritical behaviour in combination resonance area; to the influence of damping on the postcritical behaviour in panel flutter, rotor whirling and other problems of stability of elastic systems.


Dynamic Stability Parametric Resonance Instability Region Elastic System Characteristic Exponent 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bellman, R.: Stability Theory of Differential Equations, McGraw-Hill, New York 1953.MATHGoogle Scholar
  2. 2.
    Bolotin, V.V.: On the concept of stability in the mechanics of structures, in: Stability Problems in Mechanics of Structures, Stroyizdat, Moscow 1963, 6–27 (in Russian).Google Scholar
  3. 3.
    Bolotin, V.V.: Dynamic Stability of Elastic Systems, Gostekhizdat, Moscow 1956. Engl. transl.: Holden-Day, San Francisco 1963, German transi.: Verlag der Wissenshaften, Berlin 1960.Google Scholar
  4. 4.
    Schmidt, G.: Parametererregte Schwingungen, Verlag der Wissenschaften, Berlin 1975 (in German).MATHGoogle Scholar
  5. 5.
    Bolotin, V.V.: Nonconservative Problems of the Theory of Elastic Stability, Fizmatgiz, Moscow 1961 (in Russian). Engl. transi.: Pergamon Press, Oxford 1963.Google Scholar
  6. 6.
    Bolotin, V.V.: On vibrations and stability of bars subjected to nonconservatives forces, in: Vibrations in Turbomachinery, USSR Acad. Sci. Publ., Moscow 1959, 23–42 (in Russian).Google Scholar
  7. 7.
    Ziegler, H.: Linear elastic stability, Zeitschrift fr angewandte Mathematik und Physik, 4 (1953), 89–121, 167–185.CrossRefMATHGoogle Scholar
  8. 8.
    Leipholz, H.: Stability of Elastic Systems, Sijthoff and Noordhoff, Alphen an der Rijn 1980.MATHGoogle Scholar
  9. 9.
    Bisplinghoff, R.L. and H. Ashley: Principles of Aeroelasticity, Dover, New York 1962.MATHGoogle Scholar
  10. 10.
    Dowell, E.H.: A Modern Course in Aeroelasticity. Kluwer, Dordrecht 1989.CrossRefGoogle Scholar
  11. 11.
    Dowell, E.H.: Aeroelastic stability of plates and shells: an innocent’s guide to the literature, in: Instability of Continuous Systems. IUTAM Symposium Herrenalb, (Ed. H.Leipholz), Springer-Verlag, Berlin 1971, 65–77.CrossRefGoogle Scholar
  12. 12.
    Svetlitsky, V.A.: Mechanics of Cables and Flexible Rods, Mashinostroyeniye, Moscow 1978 (in Russian).Google Scholar
  13. 13.
    Dimentberg, F.M.: Bending Vibrations of Rotating Shafts. USSR Acad. Sci. Publ., Moscow 1959 (in Russian).Google Scholar
  14. 14.
    Szemplinska-Stupnicka, W.: The Behaviour of Nonlinear Vibrating Systems, 1, 2, Kluwer, Dordrecht 1990.CrossRefMATHGoogle Scholar
  15. 15.
    Lavrentyev, M. and A.Yu. Ishlinsky: Dynamic instability modes of elastic systems, Doklady AN USSR, 64 (6) (1949), 779–782 (in Russian).Google Scholar
  16. 16.
    Volmir, A.S.: Stability of Deformable Systems, Nauka, Moscow 1967 (in Russian).Google Scholar
  17. 17.
    Simitses, G.J.: Dynamic Stability of Suddenly Loaded Structures, Springer-Verlag, New York 1990.CrossRefMATHGoogle Scholar
  18. 18.
    Bolotin, V.V.: On dynamic crack propagation, Appl. Math. Mech. (PMM), 56 (1) (1990), 150–162 (in Russian).MathSciNetGoogle Scholar
  19. 19.
    Koiter, W.T.: The energy criterion of stability for continuous elastic bodies, Proc. Kon. Ned. Acad. Wet., Ser. B, 68 (1965), 178–202.MATHMathSciNetGoogle Scholar
  20. 20.
    Bolotin, V.V.: Stabilization and destabilization effects in mechanics of deformable bodies, in: Proc. 6th Canadian Congress of Applied Mechanics. CANCAM, Vancouver, 1 (1979), 1–10.Google Scholar
  21. 21.
    Meirovitch, L.: Introduction in Dynamics and Control, Wiley, New York 1985.Google Scholar
  22. 22.
    Zubov, V.I.: Stability of Motion. Lyapunov’s MethodS and Their Applications, Vysshaya Shkola, Moscow 1973 (in Russian).Google Scholar
  23. 23.
    Bolotin, V.V.: On the transverse vibrations of rods excited by periodic longitudinal forces, in: Transverse Vibrations and Critical Velocities, 1, USSR Acad. Sci. Publ., Moscow 1951, 46–77 (in Russian).Google Scholar
  24. 24.
    Nayfeh, A.H.: Problems in Perturbation, Wiley, New York 1985.MATHGoogle Scholar
  25. 25.
    Nayfeh, A.H.: Method of Normal Forms, Wiley, New York 1993.MATHGoogle Scholar
  26. 26.
    Andronov, A.A., Vitt, E.A. and S.E. Khaikin: Theory of Oscillations, Pergamon Press, Oxford 1966 (Russian original: Fizmatgiz, Moscow 1959 ).Google Scholar
  27. 27.
    Bogolyubov, N.N. and Yu.A. Mitropolsky: Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Bleash, New York 1961 (Russian original: Gostekhizdat, Moscow 1955 ).Google Scholar
  28. 28.
    Hadegorn, P.: Non-Linear Oscillations. Clarendon, Oxford 1981.Google Scholar
  29. 29.
    Nayfeh, A.H. and D.T. Mook: Nonlinear Oscillations, Wiley, New York 1979.MATHGoogle Scholar
  30. 30.
    Arnold, V.I.: Dynamical Systems, Springer-Verlag, New York 1988.MATHGoogle Scholar
  31. 31.
    Carr, J.: Applications of Centre Manifold Theory, Springer-Verlag, New York 1981.CrossRefMATHGoogle Scholar
  32. 32.
    Huseyin, K.: Nonlinear Theory of Elastic Stability. Noordhoff, Leyden 1975.MATHGoogle Scholar
  33. 33.
    Pignanataro, M., Rizzi, N. and A. Luongo, A.: Stability, Bifurcation, and Postcritical Behaviour of Elastic Systems, Elsevier, Amsterdam 1991.Google Scholar
  34. 34.
    Bazant, Z.P. and L. Cedolin: Stability of Structures. Elastic, Inelastic, Fracture and Damage Theories, Oxford University Press, New York, Oxford 1991.MATHGoogle Scholar
  35. 35.
    Arnold, V.I.: Dynamical Systems, Springer-Verlag, New York 1988.MATHGoogle Scholar
  36. 36.
    Guckenheimer, J. and P. Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York 1986.Google Scholar
  37. 37.
    Thompson, J.M.T.: Instabilities and Catastrophes in Science and Engineering, Wiley, Chichester 1982.MATHGoogle Scholar
  38. 38.
    Kounadis, A.N.: Chaoticlike and other phenomena in the nonlinear dynamic analysis of structures: quantitative-qualitative criteria, Praktika Akademias Afenon, 67 (1992), 226–276.Google Scholar
  39. 39.
    Moon, F.C.: Chaotic Vibrations: An Introduction for Applied Scientists and Engineers, Wiley, New York 1987.MATHGoogle Scholar
  40. 40.
    Moon, F.C.: Chaotic and Fractal Dynamics. Wiley, New York 1992.CrossRefGoogle Scholar
  41. 41.
    Thompson, J.M.T. and H.B. Stewart: Nonlinear Dynamics and Chaos, Wiley, Chichester 1986.MATHGoogle Scholar
  42. 42.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York 1990.CrossRefMATHGoogle Scholar
  43. 43.
    Bolotin, V.V.: On the parametric excitation of transverse vibrations, in: Transverse Vibrations and Critical Velocities, 2, USSR Acad. Sci. Publ., Moscow 1953, 5–44, 45–64 (in Russian).Google Scholar
  44. 44.
    Evan-Iwanowski, R.M.: Resonance Oscillations in Mechanical Systems, Elsevier, Amsterdam 1976.MATHGoogle Scholar
  45. 45.
    Piszczek, K.: Parametric combination resonance (of the second kind) in nonlinear systems, Rozpravy Inzh., 8 (1960), 211–229 (in Russian).MathSciNetGoogle Scholar
  46. 46.
    Nemat-Nasser, S.: On the stability of the equilibrium of nonconservative continuous systems with slight damping, J. Appl. Mech., Trans. ASME, 34 (2) (1967), 344–348.CrossRefMATHGoogle Scholar
  47. 47.
    Hagedorn, P.: On the destabilizing effect of nonlinear damping in nonconservative systems with follower forces, Int. J. Nonlinear Mech., 5 (2) (1970), 341–358.CrossRefMATHGoogle Scholar
  48. 48.
    Zhinzher, N.I.: On destabilizing effects of damping on stability of nonconservative elastic systems, Izv. USSR Academy of Sciences, Mechanics of Solids (MTT), 4 (1968), 65–68 (in Russian).Google Scholar
  49. 49.
    Bolotin, V.V. and N.I. Zhinzher: Effect of damping on stability of elastic systems subjected to nonconservative forces, Int. J. Solids and Structures, 5 (9) (1969), 965–989.CrossRefMATHGoogle Scholar
  50. 50.
    Bolotin, V.V.: Stability of viscoelastic systems subjected to nonconservative forces, in: Instability of Continuous Systems. IUTAM Symposium Herrenalb, (Ed. H.Leipholz), Springer-Verlag, Berlin 1971, 349–360.CrossRefGoogle Scholar
  51. 51.
    Bolotin, V.V., Dubovskikh, Yu.A. and N.I. Zhinzher, N.I.: Wave motion of elongated bodies subjected to nonlinear flutter, in: Nonlinear Waves in Active Medias, (Ed. J.Engelbrecht ), Springer-Verlag, Berlin 1989, 84–91CrossRefGoogle Scholar
  52. 52.
    Seyranian, A.P.: Paradox of destabilization in problems of stability of nonconservative systems, Advances in Mechanics, 13 (2) (1990), 89–124 (in Russian).MathSciNetGoogle Scholar
  53. 53.
    Novichkov, Yu.N.: Flutter of plates and shells. Advances in Science and Engineering, 11 (1978), VINITI, Moscow, 67–122 (in Russian).Google Scholar
  54. 54.
    Bolotin, V.V.: Nonlinear flutter of plates and shells, Collections in Engineering, 28 (1960), USSR Acad. Sci. Publ., 55–75, Moscow (in Russian).Google Scholar
  55. 55.
    Bolotin, V.V.: Parametric resonances in auto-oscillatory systems, Soviet Mechanics of Solids (MTT), 5 (1984), 3–10. (in Russian).MathSciNetGoogle Scholar
  56. 56.
    Bolotin, V.V.: Parametrically excited vibrations in systems with time-lag, Soviet Mechanics of Solids (MTT), 5 (1985), 14–21 (in Russian).Google Scholar
  57. 57.
    Crandall, S.H.: The physical nature of rotor instability mechanisms, Rotor Dynamical Instability. ASME Publ., AMD, 55 (1983), 1–18.MathSciNetGoogle Scholar
  58. 58.
    Schmidt, G. and A. Tondi: Non-Linear Vibrations, Akademie-Verläg, Berlin 1986.CrossRefGoogle Scholar
  59. 59.
    Poznyak, E.L.: Vibrations of rotors, in: Vibrations in Engineering, 3, (Eds. F.M.Dimentberg and K.S.Kolesnikov). Mashinostroyeniye, Moscow 1980, 130–189 (in Russian).Google Scholar
  60. 60.
    Bolotin, V.V.: Nonlinear vibrations of shafts beyond their critical speeds of rotation, in: Strength Problems in Mechanical Engineering, 1, USSR Acad. Sci. Publ., 25–53, Moscow 1958 (in Russian).Google Scholar
  61. 61.
    Dimentberg, F.M.: Transverse vibrations of a rotating beam whose cross section has different principal moments of inertia, in: Transverse Vibrations and Critical Velocities, 2, USSR Acad. Sci. Publ., Moscow 1953, 65–106 (in Russian).Google Scholar
  62. 62.
    Bolotin, V.V.: On bending vibrations of flexible shafts with cross sections of unequal stiffness, in: Collections in Engineering, 19, USSR Acad. Sci. Publ., Moscow 1954, 37–54 (in Russian).Google Scholar

Copyright information

© Springer-Verlag Wien 1995

Authors and Affiliations

  • V. V. Bolotin
    • 1
  1. 1.Russian Academy of SciencesMoscowRussia

Personalised recommendations