MOMENT LYAPUNOV EXPONENTS AND STOCHASTIC STABILITY OF A THIN-WALLED BEAM SUBJECTED TO AXIAL LOADS AND END MOMENTS

Goran Janevski, Predrag Kozić, Ratko Pavlović, Strain Posavljak

DOI Number
10.22190/FUME191127014J
First page
209
Last page
228

Abstract


In this paper, the Lyapunov exponent and moment Lyapunov exponents of two degrees-of-freedom linear systems subjected to white noise parametric excitation are investigated. The method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises. The Lyapunov exponent and moment Lyapunov exponents are important characteristics for determining both the almost-sure and the moment stability of a stochastic dynamic system. As an example, we study the almost-sure and moment stability of a thin-walled beam subjected to stochastic axial load and stochastically fluctuating end moments.  The validity of the approximate results for moment Lyapunov exponents is checked by numerical Monte Carlo simulation method for this stochastic system.

Keywords

Eigenvalues, Perturbation, Stochastic stability, Thin-walled beam, Mechanics of solids and structures

Full Text:

PDF

References


Arnold, L., Doyle, M.N., Sri Namachchivaya, N., 1997, Small noise expansion of moment Lyapunov exponents for two-dimensional systems, Dynamics and Stability of Systems, 12(3), pp. 187-211.

Khasminskii, R., Moshchuk, N., 1998, Moment Lyapunov exponent and stability index for linear conservative system with small random perturbation, SIAM Journal of Applied Mathematics, 58(1), pp. 245-256.

Sri Namachchivaya, N., Van Roessel, H.J., Talwar, S., 1994, Maximal Lyapunov exponent and almost-sure stability for coupled two-degree of freedom stochastic systems, ASME Journal of Applied Mechanics, 61, pp. 446-452.

Sri Namachchivaya, N., Van Roessel, H.J., 2004, Stochastic stability of coupled oscillators in resonance: A perturbation approach, ASME Journal of Applied Mechanics, 71, pp. 759-767.

Kozić, P., Pavlović, R., Janevski, G., 2008, Moment Lyapunov exponents of the stochastic parametrical Hill΄s equation, International Journal of Solids and Structures, 45(24), pp. 6056-6066.

Kozić, P., Janevski, G., Pavlović, R., 2009, Moment Lyapunov exponents and stochastic stability for two coupled oscillators, The Journal of Mechanics of Materials and Structures, 4(10), pp. 1689-1701.

Kozić, P., Janevski, G., Pavlović, R., 2010, Moment Lyapunov exponents and stochastic stability of a double-beam system under compressive axial load, International Journal of Solid and Structures, 47(10), pp. 1435-1442.

Milstein, N.G., Tret’Yakov, V.M., 1997, Numerical methods in the weak sense for stochastic differential equations with small noise, SIAM Journal on Numerical Analysis, 34(6), pp. 2142-2167.

Pavlović, R., Kozić, P., Rajković, P., Pavlović I., 2007, Dynamic stability of a thin-walled beam subjected to axial loads and end moments, Journal of Sound and Vibration, 301, pp. 690-700.

Xie, W.C., 2005, Monte Carlo simulation of moment Lyapunov exponents, ASME Journal of Applied Mechanics, 72, pp. 269-275.

Wedig, W., 1988, Lyapunov exponent of stochastic systems and related bifurcation problems, In: Ariaratnam, T.S., Schuëller, G.I., Elishakoff, I. (Eds.), Stochastic Structural Dynamics–Progress in Theory and Applications, Elsevier Applied Science, pp. 315 – 327.

Deng, J., Xie, W.C., Pandey M., 2014, Moment Lyapunov exponents and stochastic stability of coupled viscoelastic systems driven by white noise, Journal of mechanics of materials and structures, 9, pp. 27-50.

Deng, J., 2018, Stochastic stability of coupled viscoelastic systems excited by real noise, Mathematical problems in Engineering, Article ID 4725148.

Deng, J., Zhong, Z., Li, A., 2019, Stochastic stability of viscoelastic plates under bounded noise excitation, European Journal of Mechanics / A Solids, 78, Article ID 103849.




DOI: https://doi.org/10.22190/FUME191127014J

Refbacks

  • There are currently no refbacks.


ISSN: 0354-2025 (Print)

ISSN: 2335-0164 (Online)

COBISS.SR-ID 98732551

ZDB-ID: 2766459-4