Non-linear control of parametrically excited beams via non-collocated multi-frequency input

A non-linear control strategy is applied to a simply supported uniform beam subjected to an axial end force at the principal-parametric resonance frequency of the first skew-symmetric mode. The control input consists of the bending couples applied by two piezoceramic actuators attached onto the beam surface symmetrically with respect to the midspan and are driven by the same voltage thus generating symmetric control forces. This control architecture has zero control authority, in a linear sense, onto skew-symmetric vibrations. The non-linear transfer of energy from symmetric to skew-symmetric modes, due to non-linear inertia and curvature forces, provides the key physical mechanism for delivering effects from the actuators to the linearly uncontrollable mode. The reduced dynamics of the system, constructed with the method of multiple scales directly applied to the governing PDE's and boundary conditions, suggest effective forms of the control law as a two-frequency input. In particular, the performances of different control laws in sub-combination resonance with the excited mode are investigated. The relative phase and frequency detunings, with respect to the external excitation, are designed so as to render the control action the most effective. The control schemes generate controller forces which increase the threshold for the activation of the parametric resonance thus annihilating the parametric resonance.