Single-mode control and chaos of cantilever beam under primary and principal parametric excitations

A non-linear control law is proposed to suppress the vibrations of the first mode of a cantilever beam when subjected to primary and principal parametric excitations. The dynamics of the beam are modeled with a second-order non-linear ordinary-differential equation. The model accounts for viscous damping air drag, and inertia and geometric non-linearities. A control law based on quantic velocity feedback is proposed. The method of multiple scales method is used to derive two-first ordinary differential equations that govern the evolution of the amplitude and phase of the response. These equations are used to determine the steady state responses and their stability. Amplitude and phase modulation equations as well as external force–response and frequency–response curves are obtained. Numerical simulations confirm this scenario and detect chaos and unbounded motions in the instability regions of the periodic solutions.