Simulation and Analysis of Mechanical Systems with Parameter Fluctuation

  • Walter V. Wedig
Conference paper
Part of the IUTAM Symposia book series (IUTAM)


Mechanical systems with parameter fluctuations lead to bifurcation problems. With increasing fluctuation intensity the equilibrium of the system becomes unstable and bifurcates into non-trivial stationary solutions. To illustrate this effect we consider the typical example of a simply supported beam under axial loading. Its governing partial differential equation of motion can be reduced to a non-linear ordinary one. For the special case of white noise fluctuations we investigate Lyapunov exponents and bifurcation points of both solution forms.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Weidenhammer, F. 1969. Biegeschwingungen des Stabes unter axial pulsierender Zufallslast. VDI-Berichte Nr. 135: 101–107.Google Scholar
  2. 2.
    Khasminskii, R.Z. 1967. Necessary and sufficient conditions for asymptotic stability of linear stochastic systems. Theor. Prob. and Appls. 12: 144–147.CrossRefGoogle Scholar
  3. 3.
    Arnold, L. 1974. Stochastic Differential Equations. New York: Wiley.MATHGoogle Scholar
  4. 4.
    Wedig, W. 1988. Pitchfork and Hopf bifurcations in stochastic systems — effective methods to calculate Lyapunov exponents. To appear in: Effective Stochastic Analysis (ed. by P. Krée, W. Wedig), Heidelberg: Springer.Google Scholar
  5. 5.
    Wedig, W. 1989. Vom Chaos zur Ordnung. Gamm-Mitteilungen ISSN 0936–7195, Heft 2: 3–31.MathSciNetGoogle Scholar
  6. 6.
    Oseledec, V.I. 1968. A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19: 197–231.MathSciNetGoogle Scholar
  7. 7.
    Kozin, F. & Mitchel, R.R. 1974. Sample stability of second order linear differential equations with wide band noise coefficients. SIAM J. Appl. Math., 17: 571–605.Google Scholar
  8. 8.
    Arnold, L. & Kliemann, W. 1981. Qualitative theory of stochastic systems. In: Probabilistic Analysis and Related Topics (ed. by A.T. Bharucha-Reid). Vol. 3, New York: Academic Press.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Walter V. Wedig
    • 1
  1. 1.Institut für Technische MechanikUniversität KarlsruheKarlsruhe 1Germany

Personalised recommendations