Loss of Energy Concentration in Nonlinear Evolution Beam Equations

Abstract

In this paper, we study the combined mean field and homogenization limits for a system of weakly interacting diffusions moving in a two-scale, locally periodic confining potential, of the form considered in Duncan et al. (Brownian motion in an N-scale periodic potential, arXiv:1605.05854, 2016b). We show that, although the mean field and homogenization limits commute for finite times, they do not, in general, commute in the long time limit. In particular, the bifurcation diagrams for the stationary states can be different depending on the order with which we take the two limits. Furthermore, we construct the bifurcation diagram for the stationary McKeanVlasov equation in a two-scale potential, before passing to the homogenization limit, and we analyze the effect of the multiple local minima in the confining potential on the number and the stability of stationary solutions.

Stability analysis in some strongly prestressed rectangular plates

“…This definition extends in a quantitative way the classical linear instability to a nonlinear context, since it requires an "exponential-like" behavior of V (t) in order to fulfill condition (47); for more details, see [17,18]. Such a growth condition thus allows to highlight abrupt changes in the nature of the oscillations, from longitudinal to torsional, being more in line with the behavior of real structures.…”

“…In order to introduce our characterization of torsional instability, we recall that the vortex shedding around the deck of a bridge generates a periodic lift force f which starts the longitudinal oscillations of the structure. The frequency of f determines which longitudinal mode is prevailing [17,18] (that is, which mode captures almost all the energy from the vortices). We analyze the situation where the prevailing mode is L 1,1 .…”

“…The notion of torsional instability we deal with was introduced in [18] (see also [17]), and we here recall it. Definition 2.…”

“…In Section 3 we prove the existence of periodic (in time) solutions of (1), providing some bounds for the ones concentrated on the first longitudinal mode, which we also determine explicitly in some particular cases. In Section 4, we first analyze the stability properties of these periodic solutions and then we study the torsional stability of bi-modal solutions of (1) having one longitudinal and one torsional component; for the latter, we use a notion of stability introduced in [17,18], see Definition 2. Throughout the paper we mainly use ODE techniques, taking advantage of the results in [11,21,22,28], and numerical simulations.…”

Torsional instability and sensitivity analysis in a suspension bridge model related to the Melan equation

“…To give a precise definition in quantitative terms is a hard work. In [8] the authors give a definition of instability with respect to the time lapse considered [0, T ] with T > 0, the amplitude of oscillation and its speed of growth. We think that, choosing a proper interval [0, T ] and an appropriate rate of growth of the amplitude, such definition can be useful as a quantitative indicator of instability.…”