Nonlinear Equations for Beams and Degenerate Plates with Piers

Stability analysis in some strongly prestressed rectangular plates

“…In fact, more can be said. Since P > µ 1,1 , assuming (18) we may have multiplicity of periodic solutions, as the next statement shows. By "subharmonic solution proportional to L 1,1 " we mean a solution u of (1) in the form u(ξ, t) = z(t)L 1,1 (ξ) where, for some integer k > 1, the time coefficient z is kT -periodic, kT being its minimal period.…”

“…Theorem 3 also states that, if P > µ 1,1 , one cannot predict the long-time behavior of the solutions of (1) even for simple forces such as (18). Translated to the bridge model, this shows (2).…”

“…Theorem 3 states that if f has the form (18) and if the phase space is reduced to the first longitudinal mode, then the attractor for (1) is given by the set P of the three periodic solutions of (1) having the form (20). If in this framework it is nice to learn that there are no subharmonic solutions (called a "bad phenomenon" by Haraux [22]), on the other hand the multiplicity of periodic solutions yields an uncertainty of the behavior of all the other solutions.…”

“…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.…”

“…Fix T > 0 and let f ∈ X T be as in (18). Then there exists at least a T -periodic solution u ∈ Y T of (1) satisfying Ω u(ξ, t)e j (ξ) dξ ≡ 0 for every j ≥ 2.…”