Stability analysis in some strongly prestressed rectangular plates

Jifeng Chu; Maurizio Garrione; Filippo Gazzola

doi:10.3934/eect.2020006

Article Contents

2020, Volume 9, Issue 1: 275-299. Doi: 10.3934/eect.2020006

This issue Previous Article The Kalman condition for the boundary controllability of coupled 1-d wave equations Next Article

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

Early access: August 2019

Published: March 2020

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Abstract

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.

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Mathematics Subject Classification: 34C25, 35B35.

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Figure 1. The Deer Isle Bridge: with permission of Thaddeus Roan

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Figure 2. 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)

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Figure 3. The components $ U(t) $ (left) and $ V(t) $ (right) in (42), with (46) and $ U_0 = 0.2 $, $ V_0 = 0.01 $

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Figure 4. The components $ U(t) $ (left) and $ V(t) $ (right) in (42), with (46) and $ U_0 = 0.2 $, $ V_0 = 2 $

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Figure 5. The components $ U(t) $ (left) and $ V(t) $ (right) in (42), with (46) and $ U_0 = 0.2 $, $ V_0 = 10 $

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Figure 6. The $ V $-component of the solution of (44) for values of the parameters given by (48)

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Figure 7. The $ V $-component of the solution of (44) for values of the parameters given by (49)

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Figure 8. The $ V $-component of the solution of (44) for values of the parameters given by (50)

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