Instability of modes in a partially hinged rectangular plate

Abstract

Abstract. We consider a thin and narrow rectangular plate where the two short edges are hinged whereas the two long edges are free. This plate aims to represent the deck of a bridge, either a footbridge or a suspension bridge. We study a nonlocal evolution equation modeling the deformation of the plate and we prove existence, uniqueness and asymptotic behavior for the solutions for all initial data in suitable functional spaces. Then we prove results on the stability/instability of simple modes motivated by a phenomenon which is visible in actual bridges and we complement these theorems with some numerical experiments.

Stability analysis in some strongly prestressed rectangular plates

“…Hence, since the equation h = Γ n (h, 0) admits a unique solution, the Leray-Schauder principle guarantees the existence of a solution h ∈ (C 0 T (R)) n of h = Γ n (h, 1). This proves the existence of a T -periodic solution u n of the finite system (13). Using similar arguments as in [7,Lemmas 19 and 20] and the expression in (15), we deduce that u n (t) H 2 * and u n t (t) L 2 are bounded in [0, T ], independently of n. The equation…”

“…Back to the finite dimensional Hamiltonian system (13), this proves the claimed (C 0 T (R)) n -bound (14). Hence, since the equation h = Γ n (h, 0) admits a unique solution, the Leray-Schauder principle guarantees the existence of a solution h ∈ (C 0 T (R)) n of h = Γ n (h, 1).…”

“…We are interested in both the asymptotic stability of periodic motions and the possibility that a longitudinal oscillation suddenly transforms into a torsional oscillation. In some recent papers [4,7,13], this topic was tackled for the plate equation introduced in [14]. These papers only consider the case of weakly prestressed plates (small P ), for which the associated energy is convex, while for strongly prestressed ones (large P ) this is no longer true, although the energy remains bounded from below, see (15).…”

“…The eigenvalues and eigenfunctions of (5) were fully determined in [14] and their properties were analyzed in several subsequent works [4,7,13]. We refer to these papers for all the details, here we only recall the following facts which will be useful for our purposes.…”

Stability of a suspension bridge with a localized structural damping

“…Remark 2. In [7], the authors analyzed in detail how a solution of (1), initially oscillating in an almost purely longitudinal fashion, can suddenly start oscillating in a torsional way, even without the presence of external forces, that is, when h = 0. Therefore, we shall consider h = 0.…”