Nonlinear Rocking Motions. I: Chaos under Noisy Periodic Excitations
Publication: Journal of Engineering Mechanics
Volume 122, Issue 8
Abstract
The effects of low-intensity random perturbations on the stability of chaotic response of rocking objects under otherwise periodic excitations are examined analytically and via simulations. A stochastic Melnikov process is developed to identify a lower bound for the domain of possible chaos. An average phase-flux rate is computed to demonstrate noise effects on transitions from chaos to overturning. A mean Poincaré mapping technique is employed to reconstruct embedded chaotic attractors under random noise on Poincaré sections. Extensive simulations are employed to examine chaotic behaviors from an ensemble perspective. Analysis predicts that the presence of random perturbations enlarges the possible chaotic domain and bridges the domains of attraction of coexisting attractors. Numerical results indicate that overturning attractors are of the greatest strength among coexisting ones; and, because of the weak stability of chaotic attractors, the presence of random noise will eventually lead chaotic rocking responses to overturning. Existence of embedded strange attractors (reconstructed using mean Poincaré maps) indicates that rocking objects may experience transient chaos prior to overturn.
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References
1.
Aslam, M., Godden, W. G., and Scalise, D. T.(1980). “Earthquake rocking response of rigid bodies.”J. Struct. Engrg. Div., ASCE, 106(2), 377–392.
2.
Dimentberg, M. F., Lin, Y. K., and Zhang, R.(1993). “Toppling of computer-type equipment under base excitation.”J. Engrg. Mech. Div., ASCE, 119(1), 145–160.
3.
Frey, M., and Simiu, E.(1993). “Noise-induced chaos and phase space flux.”Physica D, 63, 321–340.
4.
Guckenheimer, J., and Holmes, P. (1983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields . Springer-Verlag, New York, N.Y.
5.
Hogan, S. J.(1989). “On the dynamics of rigid-block motion under harmonic forcing.”Proc., Royal Soc. London A, London, England, 425, 441–476.
6.
Hsieh, S.-R., Troesch, A. W., and Shaw, S. W.(1994). “Nonlinear probabilistic method for predicting vessel capsizing in random beam seas.”Proc., Royal Soc. London A, London, England, 446, 195–211.
7.
Iyengar, R. N., and Manohar, C. S.(1991). “Rocking response of rectangular rigid blocks under random noise base excitations.”Int. J. Nonlinear Mech., 26, 885–892.
8.
Kapitaniak, T. (1988). Chaos in systems with noise . World Scientific, Singapore.
9.
Koh, A. S.(1986). “Rocking of rigid blocks on randomly shaking foundations.”Nuclear Engrg. and Des., 97, 269–276.
10.
Lin, H., and Yim, S. C. S.(1996). “Nonlinear rocking motions. II: overturning under random excitations.”J. Engrg. Mech., ASCE, 122(8), 728–735.
11.
Shinozuka, M.(1971). “Simulation of multivariate and multidimensional random processes.”J. Acoustical Soc. of Am., 49, 357–367.
12.
Soong, T. T., and Grigoriu, M. (1992). Random vibration of mechanical and structural systems . Prentice-Hall, Englewood Cliffs, N.J.
13.
Spanos, P. D., and Koh, A. S.(1984). “Rocking of rigid blocks due to harmonic shaking.”J. Engrg. Mech. Div., ASCE, 110(11), 1627–1642.
14.
Stratonovich, R. L. (1967). Topics in the theory of random noise. I . Gordon and Breach, New York, N.Y.
15.
Thompson, J. M. T., and Stewart, H. B. (1986). Nonlinear dynamics and chaos . John Wiley & Sons, Chichester, England.
16.
Tso, W. K., and Wong, C. M.(1989). “Steady state rocking response of rigid blocks. Part I: analysis.”Earthquake Engrg. and Struct. Dyn., 18, 89–106.
17.
Wiggins, S. (1988). Global bifurcations and chaos: analytical methods . Springer-Verlag, New York, N.Y.
18.
Wiggins, S. (1990). Introduction to applied nonlinear dynamical systems and chaos . Springer-Verlag, New York, N.Y.
19.
Wong, C. M., and Tso, W. K.(1989). “Steady state rocking response of rigid blocks. Part II: experiment.”Earthquake Engrg. and Struct. Dyn., 18, 107–120.
20.
Yim, S. C. S., Chopra, A. K., and Penzien, J.(1980). “Rocking response of rigid blocks to earthquakes.”Earthquake Engrg. and Struct. Dyn., 8, 565–587.
21.
Yim, S. C. S., and Lin, H.(1991a). “Nonlinear impact and chaotic response of slender rocking objects.”J. Engrg. Mech. Div., ASCE, 117, 2079–2100.
22.
Yim, S. C. S., and Lin, H.(1991b). “Chaotic behavior and stability of free-standing offshore equipment.”Oc. Engrg., 18, 225–250.
23.
Yim, S. C. S., and Lin, H. (1992). “Probabilistic analysis of a chaotic dynamical system.”Applied chaos, J. H. Kim and J. Stringer, eds., John Wiley & Sons, New York, N.Y., 219–241.
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Published In
Journal of Engineering Mechanics
Volume 122 • Issue 8 • August 1996
Pages: 719 - 727
Copyright
Copyright © 1996 American Society of Civil Engineers.
History
Published online: Aug 1, 1996
Published in print: Aug 1996
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