An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip

Abstract

We study the asymptotic behavior for a system consisting of a clamped flexible beam that carries a tip payload, which is attached to a nonlinear damper and a nonlinear spring at its end. Characterizing the ω-limit sets of the trajectories, we give a sufficient condition under which the system is asymptotically stable. In the case when this condition is not satisfied, we show that the beam deflection approaches a non-decaying time-periodic solution.2010 Mathematics Subject Classification. 35B40, 70K20, 74K10, 47H20, 35Q70.

Stability Analysis of the Observer Error of an In-Domain Actuated Vibrating String

“…3.64] requires the precompactness of the solution trajectories, which is not ensured in the infinitedimensional scenario. Since in the previous section we have shown that A −1 is bounded, by means of the Sobolev embedding theorem, it follows that A −1 is compact (see proof of Lemma 2.4 in [3] or [1, p. 201]), which further implies the precompactness of the trajectories, see [2,Rem. 4.2].…”

“…Therefore, a lot of research effort has been invested in this topic, where for example the stability analysis of mechanical systems with certain boundary conditions has been addressed. For instance, in [2] the stability of an Euler-Bernoulli beam subjected to nonlinear damping and a nonlinear spring at the tip is analysed, whereas [3] is concerned with the stability behaviour of a gantry crane with heavy chain and payload. Furthermore, the proof of stability of a Lyapunov-based control law as well as a Lyapunov-based observer design for an in-domain actuated Euler-Bernoulli beam has been presented in [4].…”

Observer design for a single mast stacker crane

“…Next, regarding the investigation of the asymptotic stability of the observer error we use LaSalle's invariance principle for infinite-dimensional systems, see [17,Theorem 3.64,3.65], which can be applied since the solution trajectories are precompact. This follows from the fact that the boundedness of A −1 implies that A −1 is also compact, see [17, p. 201] and [18,Remark 4.2]. Thus, we investigate the set S = {χ ∈ X | ̇H e = 0}, where we consequently have ̇w| L = 0, and therefore, ̈w| L = 0 has to hold.…”

Stability of an Euler-Bernoulli Beam With a Nonlinear Dynamic Feedback System

“…Hence, the above result still holds true for more general operators L and N , which satisfy the mentioned properties. The proof of Lemma 5.5 is analogous to the proof of Lemma 5.4 in [22], see also [27] for a general version of this lemma. Proof.…”