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

TL;DR

This paper investigates the long-term behavior of a clamped beam with nonlinear damping and spring at its tip, identifying conditions for stability and the emergence of periodic oscillations.

Contribution

It provides a necessary stability condition for a nonlinear beam system and characterizes the asymptotic behavior, including non-decaying periodic solutions.

Findings

System stability depends on specific nonlinear damping and spring parameters.

When stability condition is not met, the beam exhibits persistent periodic oscillations.

The analysis characterizes omega-limit sets for the nonlinear beam system.

Abstract

We study the asymptotic behaviour 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 omega-limit sets of the trajectories, we give a necessary 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.