# American Institute of Mathematical Sciences

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

Received  February 2021 Revised  June 2021 Published  May 2022 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 and Continuous Dynamical Systems - S, 2022, 15 (5) : 1165-1181. 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. [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.
Function $\psi$
Smooth function $\eta.$
 [1] Ivana Bochicchio, Claudio Giorgi, Elena Vuk. On the viscoelastic coupled suspension bridge. Evolution Equations and Control Theory, 2014, 3 (3) : 373-397. doi: 10.3934/eect.2014.3.373 [2] Ling Xu, Jianhua Huang, Qiaozhen Ma. Random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021318 [3] Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control and Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251 [4] Zhong-Jie Han, Zhuangyi Liu, Jing Wang. Sharper and finer energy decay rate for an elastic string with localized Kelvin-Voigt damping. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1455-1467. doi: 10.3934/dcdss.2022031 [5] Takeshi Taniguchi. Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1571-1585. doi: 10.3934/cpaa.2017075 [6] Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147 [7] Gongwei Liu, Baowei Feng, Xinguang Yang. Longtime dynamics for a type of suspension bridge equation with past history and time delay. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4995-5013. doi: 10.3934/cpaa.2020224 [8] A. F. Almeida, M. M. Cavalcanti, J. P. Zanchetta. Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2039-2061. doi: 10.3934/cpaa.2018097 [9] Irena Lasiecka, Roberto Triggiani. Heat--structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1515-1543. doi: 10.3934/cpaa.2016001 [10] Jing Zhang. The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework. Evolution Equations and Control Theory, 2017, 6 (1) : 135-154. doi: 10.3934/eect.2017008 [11] Vo Anh Khoa, Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms. Evolution Equations and Control Theory, 2019, 8 (2) : 359-395. doi: 10.3934/eect.2019019 [12] Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115 [13] Marco Campo, José R. Fernández, Maria Grazia Naso. A dynamic problem involving a coupled suspension bridge system: Numerical analysis and computational experiments. Evolution Equations and Control Theory, 2019, 8 (3) : 489-502. doi: 10.3934/eect.2019024 [14] Suping Wang, Qiaozhen Ma. Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1299-1316. doi: 10.3934/dcdsb.2019221 [15] Quang-Minh Tran, Hong-Danh Pham. Global existence and blow-up results for a nonlinear model for a dynamic suspension bridge. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4521-4550. doi: 10.3934/dcdss.2021135 [16] Mounir Afilal, Abdelaziz Soufyane, Mauro de Lima Santos. Piezoelectric beams with magnetic effect and localized damping. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021056 [17] Peng Sun. Exponential decay of Lebesgue numbers. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3773-3785. doi: 10.3934/dcds.2012.32.3773 [18] To Fu Ma, Paulo Nicanor Seminario-Huertas. Attractors for semilinear wave equations with localized damping and external forces. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2219-2233. doi: 10.3934/cpaa.2020097 [19] Aníbal Rodríguez-Bernal, Enrique Zuazua. Parabolic singular limit of a wave equation with localized boundary damping. Discrete and Continuous Dynamical Systems, 1995, 1 (3) : 303-346. doi: 10.3934/dcds.1995.1.303 [20] Louis Tebou. Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7117-7136. doi: 10.3934/dcds.2016110

2021 Impact Factor: 1.865