# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021089
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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

Received  February 2021 Revised  June 2021 Early access August 2021

Fund Project: The second and third authors are supported by KFUPM-Project #SB201003

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.
Citation: Zayd Hajjej, Mohammad Al-Gharabli, Salim Messaoudi. Stability of a suspension bridge with a localized structural damping. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021089
##### 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.  Google Scholar [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). Google Scholar [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). Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] J.-L. Lions, Contrôlabilité exacte des systèmes distribués, Masson, Paris, 1988.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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). Google Scholar [18] M. Tucsnak, Semi-internal stabilization for a non-linear Bernoulli-Euler equation, Mathematical Methods in the Applied Sciences, 19 (1996), 897-907.   Google Scholar [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.  Google Scholar

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.  Google Scholar [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). Google Scholar [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). Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] J.-L. Lions, Contrôlabilité exacte des systèmes distribués, Masson, Paris, 1988.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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). Google Scholar [18] M. Tucsnak, Semi-internal stabilization for a non-linear Bernoulli-Euler equation, Mathematical Methods in the Applied Sciences, 19 (1996), 897-907.   Google Scholar [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.  Google Scholar
Function $\psi$
Smooth function $\eta.$
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