Loss of Energy Concentration in Nonlinear Evolution Beam Equations

Abstract

Motivated by the oscillations that were seen at the Tacoma Narrows Bridge, we introduce the notion of solutions with a prevailing mode for the nonlinear evolution beam equation

$$\begin{aligned} u_{tt} + u_{xxxx} + f(u)= g(x, t) \end{aligned}$$

in bounded space–time intervals. We give a new definition of instability for these particular solutions, based on the loss of energy concentration on their prevailing mode. We distinguish between two different forms of energy transfer, one physiological (unavoidable and depending on the nonlinearity) and one due to the insurgence of instability. We then prove a theoretical result allowing to reduce the study of this kind of infinite-dimensional stability to that of a finite-dimensional approximation. With this background, we study the occurrence of instability for three different kinds of nonlinearities f and for some forcing terms g, highlighting some of their structural properties and performing some numerical simulations.

Appendix

We state here the result that we have used in Sect. 6 for the computation of some integral terms when dealing with the nonlinearity $f(u)=u^3$. The proof follows from the prosthaphaeresis formulas.

Lemma 16

For all $p\in {\mathbb {N}}$, we have

$$\begin{aligned} \frac{8}{\pi }\int _0^\pi \sin ^4(px)\, \mathrm{d}x=3 . \end{aligned}$$

For all $p,q\in {\mathbb {N}}$ ($q\ne p$), we have

$$\begin{aligned} \frac{8}{\pi }\int _0^\pi \sin ^2(px)\sin ^2(qx)\, \mathrm{d}x=2 . \end{aligned}$$

For all $p,q\in {\mathbb {N}}$ ($q\ne p$), we have

$$\begin{aligned} \frac{8}{\pi }\int _0^\pi \sin ^3(px)\sin (qx)\, \mathrm{d}x=\left\{ \begin{array}{ll}-1\ &{}\quad \hbox {if }q=3p\\ 0\ &{}\quad \hbox {if }q\ne 3p . \end{array}\right. \end{aligned}$$

For all $p,q,r\in {\mathbb {N}}$ (all different and $q<r$), we have

$$\begin{aligned} \frac{8}{\pi }\int _0^\pi \sin ^2(px)\sin (qx)\sin (rx)\, \mathrm{d}x=\left\{ \begin{array}{ll}1\ &{}\quad \hbox {if }r+q=2p\\ -1\ &{}\quad \hbox {if }r-q=2p\\ 0\ &{}\quad \hbox {if }r\pm q\ne 2p . \end{array}\right. \end{aligned}$$

For all $p,q,r,s\in {\mathbb {N}}$ (all different and $p<q$, $r<s$), we have

$$\begin{aligned} \frac{8}{\pi }\int _0^\pi \sin (px)\sin (qx)\sin (rx)\sin (sx)\, \mathrm{d}x=\left\{ \begin{array}{ll}1\ &{}\quad \hbox {if }q\pm p=s\pm r\\ -1\ &{}\quad \hbox {if }q\pm p=s\mp r\\ 0\ &{}\quad \hbox {otherwise.} \end{array}\right. \end{aligned}$$