Figure 1.
Function $ \psi $
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May
2022, 15(5): 1165-1181.
doi: 10.3934/dcdss.2021089
Stability of a suspension bridge with a localized structural damping
| 1. | Department of Mathematics, University of Gabès, Gabès, Tunisia |
| 2. | The Preparatory Year Program and The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran, KSA |
| 3. | Department of Mathematics, College of Sciences, University of Sharjah, P.O.Box 27272, Sharjah, UAE |
Strong vibrations can cause lots of damage to structures and break materials apart. The main reason for the Tacoma Narrows Bridge collapse was the sudden transition from longitudinal to torsional oscillations caused by a resonance phenomenon. There exist evidences that several other bridges collapsed for the same reason. To overcome unwanted vibrations and prevent structures from resonating during earthquakes, winds, ..., features and modifications such as dampers are used to stabilize these bridges. In this work, we use a minimum amount of dissipation to establish exponential decay- rate estimates to the following nonlocal evolution equation
| $ u_{tt}(x,y,t)+\Delta^2 u(x,y,t) - \phi(u) u_{xx}- \left(\alpha(x, y) u_{xt}(x,y,t)\right)_x = 0, $ |
which models the deformation of the deck of either a footbridge or a suspension bridge.
Mathematics Subject Classification:
Primary: 35L51, 35L71; Secondary: 35B35, 35B41.
Citation:
Zayd Hajjej, Mohammad Al-Gharabli, Salim Messaoudi. Stability of a suspension bridge with a localized structural damping. Discrete and Continuous Dynamical Systems - S,
2022, 15
(5)
: 1165-1181.
doi: 10.3934/dcdss.2021089
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References:
| [1] |
M. Al-Gwaiz, V. Benci and F. Gazzola,
Bending and stretching energies in a rectangular plate modeling suspension bridges, Nonlinear Anal., 106 (2014), 181-734.
doi: 10.1016/j.na.2014.04.011. |
| [2] |
O. H. Ammann, T. von Karman and G. B. Woodruff, The failure of the Tacoma Narrows Bridge, Federal Works Agency, Washington D.C., (1941). |
| [3] |
F. Bleich, C. B. McCullough, R. Rosecrans and G. S. Vincent, The mathematical theory of vibration in suspension bridges, U.S. Dept. of Commerce, Bureau of Public Roads, Washington D.C., (1950). |
| [4] |
A. D. D. Cavalcanti, M. M. Cavalcanti and W. J. Corrêa et al,
Uniform decay rates for a suspension bridge with locally distributed nonlinear damping, Journal of the Franklin Institute, 357 (2020), 2388-2419.
doi: 10.1016/j.jfranklin.2020.01.004. |
| [5] |
M. M. Cavalcanti, W. J. Corrêa, R. Fukuoka and Z. Hajjej, Stabilization of a suspension bridge with locally distributed damping, Math. Control Signals Syst., 30 (2018), Art. 20, 39 pp.
doi: 10.1007/s00498-018-0226-0. |
| [6] |
A. Ferrero and F. Gazzola,
A partially hinged rectangular plate as a model for suspension bridges, Discrete Contin. Dyn. Syst. A, 35 (2015), 5879-5908.
doi: 10.3934/dcds.2015.35.5879. |
| [7] |
V. Ferreira Jr., F. Gazzola and E. Moreira dos Santos,
Instability of modes in a partially hinged rectangular plate, J. Differential Equations, 261 (2016), 6302-6340.
doi: 10.1016/j.jde.2016.08.037. |
| [8] |
F. Gazzola, Mathematical Models for Suspension Bridges: Nonlinear Structural Instability, Modeling, Simulation and Applications, 15 2015, Springer-Verlag.
doi: 10.1007/978-3-319-15434-3. |
| [9] |
J. Glover, A. C. Lazer and P. J. Mckenna,
Existence and stability of of large scale nonlinear oscillation in suspension bridges, Z. Angew. Math. Phys., 40 (1989), 172-200.
doi: 10.1007/BF00944997. |
| [10] |
Z. Hajjej and S. A. Messaoudi,
Stability of a suspension bridge with structural damping, Annales Polonici Mathematici, 125 (2020), 59-70.
doi: 10.4064/ap191023-4-2. |
| [11] |
A. C. Lazer and P. J. McKenna,
Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.
doi: 10.1137/1032120. |
| [12] |
J.-L. Lions, Contrôlabilité exacte des systèmes distribués, Masson, Paris, 1988. |
| [13] |
W. Liu and H. Zhuang, Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms, Nonlinear Differ. Equ. Appl., 24 (2017), Paper No. 67, 35 pp.
doi: 10.1007/s00030-017-0491-5. |
| [14] |
P. J. McKenna and W. Walter,
Nonlinear oscillations in a suspension bridge, Arch. Rat. Mech. Anal., 98 (1987), 167-177.
doi: 10.1007/BF00251232. |
| [15] |
S. A. Messaoudi and S. E. Mukiawa, A suspension bridge problem: Existence and stability, Mathematics Across Contemporary Sciences, 2017,151–165.
doi: 10.1007/978-3-319-46310-0_9. |
| [16] |
S. A. Messaoudi and S. E. Mukiawa,
Existence and stability of fourth-order nonlinear plate problem, Nonauton. Dyn. Syst., 6 (2019), 81-98.
doi: 10.1515/msds-2019-0006. |
| [17] |
F. C. Smith and G. S. Vincent, Aerodynamic stability of suspension bridges: With special reference to the Tacoma Narrows Bridge, Part Ⅱ: Mathematical analysis, Investigation conducted by the Structural Research Laboratory, University of Washington, University of Washington Press, Seattle, (1950). |
| [18] |
M. Tucsnak,
Semi-internal stabilization for a non-linear Bernoulli-Euler equation, Mathematical Methods in the Applied Sciences, 19 (1996), 897-907.
|
| [19] |
Y. Wang,
Finite time blow-up and global solutions for fourth-order damped wave equations, Journal of Mathematical Analysis and Applications, 418 (2014), 713-733.
doi: 10.1016/j.jmaa.2014.04.015. |
show all references
References:
| [1] |
M. Al-Gwaiz, V. Benci and F. Gazzola,
Bending and stretching energies in a rectangular plate modeling suspension bridges, Nonlinear Anal., 106 (2014), 181-734.
doi: 10.1016/j.na.2014.04.011. |
| [2] |
O. H. Ammann, T. von Karman and G. B. Woodruff, The failure of the Tacoma Narrows Bridge, Federal Works Agency, Washington D.C., (1941). |
| [3] |
F. Bleich, C. B. McCullough, R. Rosecrans and G. S. Vincent, The mathematical theory of vibration in suspension bridges, U.S. Dept. of Commerce, Bureau of Public Roads, Washington D.C., (1950). |
| [4] |
A. D. D. Cavalcanti, M. M. Cavalcanti and W. J. Corrêa et al,
Uniform decay rates for a suspension bridge with locally distributed nonlinear damping, Journal of the Franklin Institute, 357 (2020), 2388-2419.
doi: 10.1016/j.jfranklin.2020.01.004. |
| [5] |
M. M. Cavalcanti, W. J. Corrêa, R. Fukuoka and Z. Hajjej, Stabilization of a suspension bridge with locally distributed damping, Math. Control Signals Syst., 30 (2018), Art. 20, 39 pp.
doi: 10.1007/s00498-018-0226-0. |
| [6] |
A. Ferrero and F. Gazzola,
A partially hinged rectangular plate as a model for suspension bridges, Discrete Contin. Dyn. Syst. A, 35 (2015), 5879-5908.
doi: 10.3934/dcds.2015.35.5879. |
| [7] |
V. Ferreira Jr., F. Gazzola and E. Moreira dos Santos,
Instability of modes in a partially hinged rectangular plate, J. Differential Equations, 261 (2016), 6302-6340.
doi: 10.1016/j.jde.2016.08.037. |
| [8] |
F. Gazzola, Mathematical Models for Suspension Bridges: Nonlinear Structural Instability, Modeling, Simulation and Applications, 15 2015, Springer-Verlag.
doi: 10.1007/978-3-319-15434-3. |
| [9] |
J. Glover, A. C. Lazer and P. J. Mckenna,
Existence and stability of of large scale nonlinear oscillation in suspension bridges, Z. Angew. Math. Phys., 40 (1989), 172-200.
doi: 10.1007/BF00944997. |
| [10] |
Z. Hajjej and S. A. Messaoudi,
Stability of a suspension bridge with structural damping, Annales Polonici Mathematici, 125 (2020), 59-70.
doi: 10.4064/ap191023-4-2. |
| [11] |
A. C. Lazer and P. J. McKenna,
Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.
doi: 10.1137/1032120. |
| [12] |
J.-L. Lions, Contrôlabilité exacte des systèmes distribués, Masson, Paris, 1988. |
| [13] |
W. Liu and H. Zhuang, Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms, Nonlinear Differ. Equ. Appl., 24 (2017), Paper No. 67, 35 pp.
doi: 10.1007/s00030-017-0491-5. |
| [14] |
P. J. McKenna and W. Walter,
Nonlinear oscillations in a suspension bridge, Arch. Rat. Mech. Anal., 98 (1987), 167-177.
doi: 10.1007/BF00251232. |
| [15] |
S. A. Messaoudi and S. E. Mukiawa, A suspension bridge problem: Existence and stability, Mathematics Across Contemporary Sciences, 2017,151–165.
doi: 10.1007/978-3-319-46310-0_9. |
| [16] |
S. A. Messaoudi and S. E. Mukiawa,
Existence and stability of fourth-order nonlinear plate problem, Nonauton. Dyn. Syst., 6 (2019), 81-98.
doi: 10.1515/msds-2019-0006. |
| [17] |
F. C. Smith and G. S. Vincent, Aerodynamic stability of suspension bridges: With special reference to the Tacoma Narrows Bridge, Part Ⅱ: Mathematical analysis, Investigation conducted by the Structural Research Laboratory, University of Washington, University of Washington Press, Seattle, (1950). |
| [18] |
M. Tucsnak,
Semi-internal stabilization for a non-linear Bernoulli-Euler equation, Mathematical Methods in the Applied Sciences, 19 (1996), 897-907.
|
| [19] |
Y. Wang,
Finite time blow-up and global solutions for fourth-order damped wave equations, Journal of Mathematical Analysis and Applications, 418 (2014), 713-733.
doi: 10.1016/j.jmaa.2014.04.015. |
Figure 2.
Smooth function $ \eta. $
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