Nonlinear Loading and Damping of a Single Degree of Freedom Oscillator

A single degree of freedom oscillator system subject to regular sinusoidal loading does exhibit a well-known linear solution. In real situations, we shall add damping as there in any physical system will be a damping mechanism that will limit the oscillations. In case of forced oscillations in a fluid, the loading and the damping may be nonlinear and the response may then exhibit nonlinear characteristics. A simple drag loading term proportional to the square of the oscillator's velocity is studied as an example of a system exhibiting limit cycles: When the external loading and the oscillator are in higher order resonances, where the natural period of the loading is an integer of the natural period of the system, resonances will occur. The resulting response of the system is a limit cycle oscillation. Furthermore, for oscillators with non-negligible motions, the motion will act as a nonlinear damping term, that for some situations may be interpreted as "negative damping." In case of irregular wave loading, the system can get into higher order resonances occasionally (when the periods of the main waves are an integer of the oscillator's periods), triggering large temporary limit cycle motions. A review of other nonlinear damping models is also given with a brief discussion of the performance of systems exposed to these damping models.