zbMATH — the first resource for mathematics

Erosion of the safe basin for the transversal oscillations of a suspension bridge. (English) Zbl 1129.74317
Summary: The time evolution of the lowest order transversal oscillation mode of a suspension bridge is studied by means of a piecewise-linear forced and damped one-dimensional oscillator, in which the loss of smoothness is due to the asymmetric response of the bridge hangers with respect to stretching and compression. If the midpoint roadbed deflection is outside a specified safe region, the bridge is supposed to collapse. We analyze the relative area of the safe basin, or the fraction of initial conditions in the phase space for which the bridge does not collapse with respect to the damping and forcing parameters. The safe basin erosion is enhanced by the appearance of incursive fingers caused by the exponential accumulation of safe basin lobes towards an invariant manifold of a periodic orbit which undergoes a homoclinic bifurcation.

74H45 Vibrations in dynamical problems in solid mechanics
74H60 Dynamical bifurcation of solutions to dynamical problems in solid mechanics
Full Text: DOI
[1] Billah, K.Y.; Scanlan, R.H., Resonance, tacoma narrows bridge failure and undergraduate physics textbooks, Am. J. phys., 59, 118-124, (1991)
[2] Lazer, A.C.; McKenna, P.J., Large amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM rev., 58, 537-578, (1990) · Zbl 0725.73057
[3] Images of the disaster are available in the website http://cee.carleton.ca/Exhibits/TacomaNarrows/, taken from the 20-minute silent movie
[4] Nayfeh, A.H.; Mook, D.T., Nonlinear oscillations, (1979), Wiley-Interscience New York
[5] Shaw, S.W.; Holmes, P.J., A periodically forced piecewise linear oscillator, J. sound vibrat., 90, 29-155, (1983) · Zbl 0561.70022
[6] Kim, Y.B.; Noah, S.T., Stability and bifurcation analysis of oscillators with piecewise-linear characteristics: a general approach, J. appl. mech., 58, 545-553, (1991) · Zbl 0850.70235
[7] Blazejczyk-Okulewska, B.; Czolczynski, K.; Kapitaniak, T.; Wojewoda, J., Chaotic mechanics in systems with friction and impacts, (1999), World Scientific Singapore · Zbl 0933.70002
[8] Blazejczyk-Okulewska, B., Study of the impact oscillator with elastic coupling of masses, Chaos, solitons & fractals, 11, 2487-2492, (2000) · Zbl 0955.70506
[9] Wiercigroch, M.; DeKraker, B., Applied nonlinear dynamics and chaos of mechanical systems with discontinuities, (2000), World Scientific Singapore
[10] Wiercigroch, M., Modelling of dynamical systems with motion dependent discontinuities, Chaos, solitons & fractals, 11, 2429-2442, (2000) · Zbl 0966.70017
[11] Doole, S.H.; Hogan, S.J., A piecewise linear suspension bridge model: nonlinear dynamics and orbit continuation, Dynam. stab. syst., 11, 19-47, (1996) · Zbl 0855.34041
[12] Freitas MST, Viana RL, Grebogi C. Multistability, basin boundary structure, and chaotic behavior in a suspension bridge model. Int J Bif Chaos, in press · Zbl 1129.74318
[13] Thompson, J.M.T.; Rainey, R.C.T.; Soliman, M.S., Ship stability criteria based on chaotic transients from incursive fractals, Phil. trans. R. soc. lond. A, 332, 149-167, (1990) · Zbl 0709.76021
[14] Hindmarch, A.C., ODEPACK: a systematized collection of ODE solvers, ()
[15] Soliman, M.S.; Thompson, J.M.T., Global dynamics underlying sharp basin erosion in nonlinear driven oscillators, Phys. rev. A, 45, 3425-3431, (1992)
[16] Soliman, M.S., Fractal erosion of basins of attraction in coupled nonlinear systems, J. sound vib., 182, 727-740, (1995) · Zbl 1237.70089
[17] Kükner, A., A view on capsizing under direct and parametric wave excitation, Ocean engng., 25, 677-685, (1998)
[18] Senjanovic, I.; Parunov, J.; Cipric, G., Safety analysis of ship rolling in rough sea, Chaos, solitons & fractals, 4, 659-680, (1997) · Zbl 0963.70541
[19] McDonald, S.W.; Grebogi, C.; Ott, E.; Yorke, J.A., Fractal basin boundaries, Physica D, 17, 125-153, (1985) · Zbl 0588.58033
[20] Moon, F.C.; Li, G.-.X., Fractal basin boundaries and homoclinic orbits for periodic motion in a two-well potential, Phys. rev. lett., 55, 1439-1442, (1985)
[21] Pentek, A.; Toroczkai, Z.; Tel, T.; Grebogi, C.; Yorke, J.A., Fractal boundaries in open hydrodynamical flows: signatures of chaotic saddles, Phys. rev. E, 51, 4076-4088, (1995)
[22] Doole, S.H.; Hogan, S.J., Nonlinear dynamics of the extended lazer – mckenna bridge oscillation model, Dynam. stab. syst., 15, 43-58, (2000) · Zbl 0952.34042
[23] Feudel, U.; Grebogi, C., Multistability and the control of complexity, Chaos, solitons & fractals, 7, 597-604, (1997) · Zbl 0933.37032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.