Non-linear cancellation of the parametric resonance in elastic beams: Theory and experiment

A non-linear control strategy is applied to a simply supported uniform elastic 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 pairs of piezoceramic actuators attached onto both sides of the beam surfaces and symmetrically with respect to the midspan, driven by the same voltage, thus resulting into 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 motions to skew-symmetric modes, due to non-linear inertia and curvature effects, provides the key physical mechanism for channelling suitable control power from the actuators into 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 sub-combination resonance with the parametrically driven mode. The performances of different control laws are investigated. The relative phase and frequency relationships are designed so as to render the control action the most effective. The control schemes generate non-linear controller forces which increase the threshold for the activation of the parametric resonance thus resulting into its annihilation. The theoretical predictions are compared with experimentally obtained results.