Oscillations of a nonlinearly damped extensible beam

The Nonlinearly Damped Oscillator

We study the large-time behaviour of the nonlinear oscillator mx′′ + f(x′) + k x = 0 , where m, k > 0 and f is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case f(x′) = A |x′|α−1x′ with α real, A > 0. We characterize the existence and behaviour of fast orbits, i....

Exponential decay of solutions of a nonlinearly damped wave equation

The issue of stablity of solutions to nonlinear wave equations has been addressed by many authors. So many results concerning energy decay have been established. Here in this paper we consider the following nonlinearly damped wave equation utt −∆u+ a(1 + |ut|)ut = bu|u|p−2, a, b > 0, in a bounded domain and show that, for suitably chosen initial data, the energy of the solution decays exponenti...

Blow Up in a Nonlinearly Damped Wave Equation

In this paper we consider the nonlinearly damped semilinear wave equation utt −∆u + aut |ut|m−2 = bu|u|p−2 associated with initial and Dirichlet boundary conditions. We prove that any strong solution, with negative initial energy, blows up in finite time if p > m. This result improves an earlier one in [2].

Global Nonexistence in a Nonlinearly Damped Wave Equation

Oscillations of a quadratically damped pendulum

A physical pendulum consisting of a circular disk at the end of a thin metal rod is connected to a low-friction rotary motion sensor, so that its angular position and velocity can be accurately measured. The disk can be oriented either perpendicular or parallel to the plane of swing to give significant or negligible air drag, respectively. The motion is analytically modeled in phase space. A qu...