Suppression of Vibrations of a Forced and Self-excited Nonlinear Beam by Using Positive Position Feedback Controller PPF

This paper presents a quantitative analysis on the nonlinear behavior of a forced and self-excited beam coupled with a positive position feedback controller PPF. Such that the external excitation is a harmonic motion on the support of the cantilever beam. Self-excitation is caused by fluid flow and modelled by a nonlinear damping with a negative linear part (Rayleigh’s function). Self-excitation can build up oscillations even in the absence of external forces. Also self-excitation can interact with the external excitation and lead system to vibrate with a quasi-periodic motion and to be unstable. This problem is treated here by using PPF controller. It is assumed that the beam vibrates in the presence of external harmonic excitation close to its natural frequency and one to one internal resonance. Multiple scales perturbation technique MSPT is used to get a first order approximate solution of this system. The stability of the steady state solution is investigated by using the frequency-response equations. The effects of different controller parameters on beam vibrations are studied and optimum conditions for system operation are deduced. Finally, all analytical results are validated by using numerical solution.