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Research Papers

# Moment Lyapunov Exponents and Stochastic Stability of a Three-Dimensional System on Elastic Foundation Using a Perturbation Approach

[+] Author and Article Information

Department of Mechanical Engineering,
University of Niš,
Medvedeva 14,
18000 Niš,

Marko Petković

Department of Science and Mathematics,
University of Niš,
18000 Niš, Serbia

1Corresponding author.

Manuscript received May 6, 2012; final manuscript received December 29, 2012; accepted manuscript posted January 29, 2013; published online July 12, 2013. Assoc. Editor: Wei-Chau Xie.

J. Appl. Mech 80(5), 051009 (Jul 12, 2013) (10 pages) Paper No: JAM-12-1185; doi: 10.1115/1.4023519 History: Received May 06, 2012; Revised December 29, 2012; Accepted January 29, 2013

## Abstract

In this paper, the stochastic stability of the three elastically connected Euler beams on elastic foundation is studied. The model is given as three coupled oscillators. Stochastic stability conditions are expressed by the Lyapunov exponent and moment Lyapunov exponents. It is determined that the new set of transformation for getting $Ito∧$ differential equations can be applied for any system of three coupled oscillators. The method of regular perturbation is used to determine the asymptotic expressions for these exponents in the presence of small intensity noises. Analytical results are presented for the almost sure and moment stability of a stochastic dynamical system. The results are applied to study the moment stability of the complex structure with influence of the white noise excitation due to the axial compressive stochastic load.

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## References

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## Figures

Fig. 1

Geometry of complex three-beam system on elastic foundation

Fig. 2

Moment Lyapunov exponent Λ(p) for σ= 200×103,d1 = d2 = c0 = 0.01; thin lines—double beam [13], thick lines—three beam system on elastic foundation, dashed lines—second perturbation of the three beam system

Fig. 3

Stability regions for almost-sure (a-s) and pth moment stability for ɛ=0.002

Fig. 4

Factor form

Fig. 5

Solving integral

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