# American Institute of Mathematical Sciences

March  2020, 9(1): 275-299. doi: 10.3934/eect.2020006

## Stability analysis in some strongly prestressed rectangular plates

 1 Maths Department, Shanghai Normal University, Shanghai 200234, China 2 Maths Department, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy

Received  December 2018 Published  March 2020 Early access  August 2019

We consider an evolution plate equation aiming to model the motion of the deck of a periodically forced strongly prestressed suspension bridge. Using the prestress assumption, we show the appearance of multiple time-periodic uni-modal longitudinal solutions and we discuss their stability. Then, we investigate how these solutions exchange energy with a torsional mode. Although the problem is forced, we find a portrait where stability and instability regions alternate. The techniques used rely on ODE analysis of stability and are complemented with numerical simulations.

Citation: Jifeng Chu, Maurizio Garrione, Filippo Gazzola. Stability analysis in some strongly prestressed rectangular plates. Evolution Equations and Control Theory, 2020, 9 (1) : 275-299. doi: 10.3934/eect.2020006
##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series 55, Washington, D. C., 1964. [2] O. H. Ammann, T. von Kármán and G. B. Woodruff, The Failure of the Tacoma Narrows Bridge, Technical Report, Federal Works Agency. Washington, D. C., 1941. [3] U. Battisti, E. Berchio, A. Ferrero and F. Gazzola, Energy transfer between modes in a nonlinear beam equation, J. Math. Pures Appl., 108 (2017), 885-917.  doi: 10.1016/j.matpur.2017.05.010. [4] E. Berchio, A. Ferrero and F. Gazzola, Structural instability of nonlinear plates modelling suspension bridges: Mathematical answers to some long-standing questions, Nonlinear Analysis Real World Applications, 28 (2016), 91-125.  doi: 10.1016/j.nonrwa.2015.09.005. [5] H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. [6] K. Y. Billah and R. H. Scanlan, Resonance, Tacoma Narrows Bridge failure, and undergraduate physics textbooks, Amer. J. Physics, 59 (1991), 118-124. [7] D. Bonheure, F. Gazzola and E. Moreira dos Santos, Periodic solutions and torsional instability in a nonlinear nonlocal plate equation, to appear in SIAM J. Math. Anal., 51 (2019), 3052–3091. doi: 10.1137/18M1221242. [8] D. Burgreen, Free vibrations of a pin-ended column with constant distance between pin ends, J. Appl. Mech., 18 (1951), 135-139. [9] E. I. Butikov, Subharmonic resonances of the parametrically driven pendulum, J. Phys. A: Math. Gen., 35 (2002), 6209-6231. [10] I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, AKTA, Kharkiv, 1999. [11] E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions, J. Dyn. Diff. Equat., 6 (1994), 631-637.  doi: 10.1007/BF02218851. [12] E. H. Dowell, Flutter of a buckled plate as an example of chaotic motion of a deterministic autonomous system, J. Sound Vibration, 85 (1982), 333-344.  doi: 10.1016/0022-460X(82)90259-0. [13] V. J. Ferreira, F. Gazzola and E. Moreira dos Santos, Instability of modes in a partially hinged rectangular plate, J. Diff. Eq., 261 (2016), 6302-6340.  doi: 10.1016/j.jde.2016.08.037. [14] A. Ferrero and F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Discrete Contin. Dyn. Syst., 35 (2015), 5879-5908.  doi: 10.3934/dcds.2015.35.5879. [15] C. Fitouri and A. Haraux, Boundedness and stability for the damped and forced single well Duffing equation, Discrete Contin. Dyn. Syst., 33 (2013), 211-223.  doi: 10.3934/dcds.2013.33.211. [16] C. Fitouri and A. Haraux, Sharp estimates of bounded solutions to some semilinear second order dissipative equations, J. Math. Pures Appl., 92 (2009), 313-321.  doi: 10.1016/j.matpur.2009.04.010. [17] M. Garrione and F. Gazzola, Loss of energy concentration in nonlinear evolution beam equations, J. Nonlinear Science, 27 (2017), 1789-1827.  doi: 10.1007/s00332-017-9386-1. [18] M. Garrione and F. Gazzola, Nonlinear equations and stability for beams and degenerate plates with several intermediate piers, SpringerBriefs in Mathematics, Springer, Cham, 2019. [19] S. Gasmi and A. Haraux, N-cyclic functions and multiple subharmonic solutions of Duffing's equation, J. Math. Pures Appl., 97 (2012), 411-423.  doi: 10.1016/j.matpur.2009.08.005. [20] F. Gazzola, Mathematical Models for Suspension Bridges, Modeling, Simulation and Applications 15, Springer, Cham, 2015. doi: 10.1007/978-3-319-15434-3. [21] M. Ghisi, M. Gobbino and A. Haraux, An infinite dimensional Duffing-like evolution equation with linear dissipation and an asymptotically small source term, Nonlinear Anal. Real World Appl., 43 (2018), 167-191.  doi: 10.1016/j.nonrwa.2018.02.007. [22] A. Haraux, On the double well Duffing equation with a small bounded forcing term, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 29 (2005), 207-230. [23] G. Herrmann and W. Hauger, On the interrelation of divergence, flutter and auto-parametric resonance, Ingenieur-Archiv, 42 (1973), 81-88. [24] J. S. Howell, I. Lasiecka and J. T. Webster, Quasi-stability and exponential attractors for a non-gradient system: Applications to piston-theoretic plates with internal damping, Evol. Equ. Control Theory, 5 (2016), 567-603.  doi: 10.3934/eect.2016020. [25] A. Jenkins, Self-oscillation, Physics Reports, 525 (2013), 167-222.  doi: 10.1016/j.physrep.2012.10.007. [26] G. H. Knightly and D. Sather, Nonlinear buckled states of rectangular plates, Arch. Ration. Mech. Anal., 54 (1974), 356-372.  doi: 10.1007/BF00249196. [27] E. Mathieu, Mémoire sur le mouvement vibratoire d'une membrane de forme elliptique, J. Math. Pures Appl., 13 (1868), 137-203. [28] F. I. Njoku and P. Omari, Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions, Appl. Math. Comput., 135 (2003), 471-490.  doi: 10.1016/S0096-3003(02)00062-0. [29] R. Ortega, Topological degree and stability of periodic solutions for certain differential equations, J. London Math. Soc., 42 (1990), 505-516.  doi: 10.1112/jlms/s2-42.3.505. [30] M. P. Païdoussis, S. J. Price and E. de Langre, Fluid-structure Interactions. Cross-Flow-Induced Instabilities, Cambridge University Press, Cambridge, 2011. [31] R. H. Scanlan, The action of flexible bridges under wind, Ⅰ: Flutter theory, Ⅱ: Buffeting theory, J. Sound and Vibration, 60 (1978), 187–199 & 201–211. [32] R. Scott, In the Wake of Tacoma. Suspension Bridges and the Quest for Aerodynamic Stability, ASCE, Reston, 2001. [33] Tacoma Narrows Bridge collapse, (video), 1940, http://www.youtube.com/watch?v=3mclp9QmCGs. [34] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [35] S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. [36] V. A. Yakubovich and V. M. Starzhinskii, Linear differential equations with periodic coefficients, Russian Original in Izdat, Nauka, Moscow, J. Wiley & Sons, New York, 1 (1975), xii+386 pp.; 2 (1975), 387–839.

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##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series 55, Washington, D. C., 1964. [2] O. H. Ammann, T. von Kármán and G. B. Woodruff, The Failure of the Tacoma Narrows Bridge, Technical Report, Federal Works Agency. Washington, D. C., 1941. [3] U. Battisti, E. Berchio, A. Ferrero and F. Gazzola, Energy transfer between modes in a nonlinear beam equation, J. Math. Pures Appl., 108 (2017), 885-917.  doi: 10.1016/j.matpur.2017.05.010. [4] E. Berchio, A. Ferrero and F. Gazzola, Structural instability of nonlinear plates modelling suspension bridges: Mathematical answers to some long-standing questions, Nonlinear Analysis Real World Applications, 28 (2016), 91-125.  doi: 10.1016/j.nonrwa.2015.09.005. [5] H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. [6] K. Y. Billah and R. H. Scanlan, Resonance, Tacoma Narrows Bridge failure, and undergraduate physics textbooks, Amer. J. Physics, 59 (1991), 118-124. [7] D. Bonheure, F. Gazzola and E. Moreira dos Santos, Periodic solutions and torsional instability in a nonlinear nonlocal plate equation, to appear in SIAM J. Math. Anal., 51 (2019), 3052–3091. doi: 10.1137/18M1221242. [8] D. Burgreen, Free vibrations of a pin-ended column with constant distance between pin ends, J. Appl. Mech., 18 (1951), 135-139. [9] E. I. Butikov, Subharmonic resonances of the parametrically driven pendulum, J. Phys. A: Math. Gen., 35 (2002), 6209-6231. [10] I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, AKTA, Kharkiv, 1999. [11] E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions, J. Dyn. Diff. Equat., 6 (1994), 631-637.  doi: 10.1007/BF02218851. [12] E. H. Dowell, Flutter of a buckled plate as an example of chaotic motion of a deterministic autonomous system, J. Sound Vibration, 85 (1982), 333-344.  doi: 10.1016/0022-460X(82)90259-0. [13] V. J. Ferreira, F. Gazzola and E. Moreira dos Santos, Instability of modes in a partially hinged rectangular plate, J. Diff. Eq., 261 (2016), 6302-6340.  doi: 10.1016/j.jde.2016.08.037. [14] A. Ferrero and F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Discrete Contin. Dyn. Syst., 35 (2015), 5879-5908.  doi: 10.3934/dcds.2015.35.5879. [15] C. Fitouri and A. Haraux, Boundedness and stability for the damped and forced single well Duffing equation, Discrete Contin. Dyn. Syst., 33 (2013), 211-223.  doi: 10.3934/dcds.2013.33.211. [16] C. Fitouri and A. Haraux, Sharp estimates of bounded solutions to some semilinear second order dissipative equations, J. Math. Pures Appl., 92 (2009), 313-321.  doi: 10.1016/j.matpur.2009.04.010. [17] M. Garrione and F. Gazzola, Loss of energy concentration in nonlinear evolution beam equations, J. Nonlinear Science, 27 (2017), 1789-1827.  doi: 10.1007/s00332-017-9386-1. [18] M. Garrione and F. Gazzola, Nonlinear equations and stability for beams and degenerate plates with several intermediate piers, SpringerBriefs in Mathematics, Springer, Cham, 2019. [19] S. Gasmi and A. Haraux, N-cyclic functions and multiple subharmonic solutions of Duffing's equation, J. Math. Pures Appl., 97 (2012), 411-423.  doi: 10.1016/j.matpur.2009.08.005. [20] F. Gazzola, Mathematical Models for Suspension Bridges, Modeling, Simulation and Applications 15, Springer, Cham, 2015. doi: 10.1007/978-3-319-15434-3. [21] M. Ghisi, M. Gobbino and A. Haraux, An infinite dimensional Duffing-like evolution equation with linear dissipation and an asymptotically small source term, Nonlinear Anal. Real World Appl., 43 (2018), 167-191.  doi: 10.1016/j.nonrwa.2018.02.007. [22] A. Haraux, On the double well Duffing equation with a small bounded forcing term, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 29 (2005), 207-230. [23] G. Herrmann and W. Hauger, On the interrelation of divergence, flutter and auto-parametric resonance, Ingenieur-Archiv, 42 (1973), 81-88. [24] J. S. Howell, I. Lasiecka and J. T. Webster, Quasi-stability and exponential attractors for a non-gradient system: Applications to piston-theoretic plates with internal damping, Evol. Equ. Control Theory, 5 (2016), 567-603.  doi: 10.3934/eect.2016020. [25] A. Jenkins, Self-oscillation, Physics Reports, 525 (2013), 167-222.  doi: 10.1016/j.physrep.2012.10.007. [26] G. H. Knightly and D. Sather, Nonlinear buckled states of rectangular plates, Arch. Ration. Mech. Anal., 54 (1974), 356-372.  doi: 10.1007/BF00249196. [27] E. Mathieu, Mémoire sur le mouvement vibratoire d'une membrane de forme elliptique, J. Math. Pures Appl., 13 (1868), 137-203. [28] F. I. Njoku and P. Omari, Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions, Appl. Math. Comput., 135 (2003), 471-490.  doi: 10.1016/S0096-3003(02)00062-0. [29] R. Ortega, Topological degree and stability of periodic solutions for certain differential equations, J. London Math. Soc., 42 (1990), 505-516.  doi: 10.1112/jlms/s2-42.3.505. [30] M. P. Païdoussis, S. J. Price and E. de Langre, Fluid-structure Interactions. Cross-Flow-Induced Instabilities, Cambridge University Press, Cambridge, 2011. [31] R. H. Scanlan, The action of flexible bridges under wind, Ⅰ: Flutter theory, Ⅱ: Buffeting theory, J. Sound and Vibration, 60 (1978), 187–199 & 201–211. [32] R. Scott, In the Wake of Tacoma. Suspension Bridges and the Quest for Aerodynamic Stability, ASCE, Reston, 2001. [33] Tacoma Narrows Bridge collapse, (video), 1940, http://www.youtube.com/watch?v=3mclp9QmCGs. [34] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [35] S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. [36] V. A. Yakubovich and V. M. Starzhinskii, Linear differential equations with periodic coefficients, Russian Original in Izdat, Nauka, Moscow, J. Wiley & Sons, New York, 1 (1975), xii+386 pp.; 2 (1975), 387–839.
The Deer Isle Bridge: with permission of Thaddeus Roan
The choice of the lower and upper solutions in the proof of the bounds and of the stability properties of $\wp^1$ (left) and $\wp^2, \wp^3$ (right)
The components $U(t)$ (left) and $V(t)$ (right) in (42), with (46) and $U_0 = 0.2$, $V_0 = 0.01$
The components $U(t)$ (left) and $V(t)$ (right) in (42), with (46) and $U_0 = 0.2$, $V_0 = 2$
The components $U(t)$ (left) and $V(t)$ (right) in (42), with (46) and $U_0 = 0.2$, $V_0 = 10$
The $V$-component of the solution of (44) for values of the parameters given by (48)
The $V$-component of the solution of (44) for values of the parameters given by (49)
The $V$-component of the solution of (44) for values of the parameters given by (50)
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