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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 |
$ 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, $ |
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. |


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