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 utt + uxxxx + f(u)= g(x, t) 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.

Garrione, M., Gazzola, F. (2017). Loss of Energy Concentration in Nonlinear Evolution Beam Equations. JOURNAL OF NONLINEAR SCIENCE, 27(6), 1789-1827 [10.1007/s00332-017-9386-1].

Loss of Energy Concentration in Nonlinear Evolution Beam Equations

GARRIONE, MAURIZIO;
2017

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 utt + uxxxx + f(u)= g(x, t) 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.
Articolo in rivista - Articolo scientifico
Beam equation; Galerkin approximation; Initial-boundary value problems; Stability;
Initial-boundary value problems; beam equation; stability; Galerkin approximation
English
1789
1827
39
Garrione, M., Gazzola, F. (2017). Loss of Energy Concentration in Nonlinear Evolution Beam Equations. JOURNAL OF NONLINEAR SCIENCE, 27(6), 1789-1827 [10.1007/s00332-017-9386-1].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/156322
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