Open-Loop Nonlinear Vibration Control of Shallow Arches via Perturbation Approach

An open-loop nonlinear control strategy applied to a hinged-hinged shallow arch, subjected to a longitudinal end-displacement with frequency twice the frequency of the second mode (principal parametric resonance), is developed. The control action—a transverse point force at the midspan—is typical of many single-input control systems; the control authority onto part of the system dynamics is high whereas the control authority onto some other part of the system dynamics is zero within the linear regime. However, although the action of the controller is orthogonal, in a linear sense, to the externally excited first antisymmetric mode, beneficial effects are exerted through nonlinear actuator action due to the system structural nonlinearities. The employed mechanism generating the effective nonlinear controller action is a one-half subharmonic resonance (control frequency being twice the frequency of the excited mode). The appropriate form of the control signal and associated phase is suggested by the dynamics at reduced orders, determined by a multiple-scales perturbation analysis directly applied to the integral-partial-differential equations of motion and boundary conditions. For optimal control phase and gain—the latter obtained via a combined analytical and numerical approach with minimization of a suitable cost functional—the parametric resonance is cancelled and the response of the system is reduced by orders of magnitude near resonance. The robustness of the proposed control methodology with respect to phase and frequency variations is also demonstrated.