Simulating Active Vibration Attenuation in Underactuated Spatial Structures

The active optimal attenuation of vibrations in spatial structures, modeled by finite elements with a possibly large number of degrees of freedom, is considered. The problem is formulated as underactuated in modal space, such that the desired number of controlled modes may be greater than the number of independent discrete actuators. Consequently, modal variables are coupled via second-order nonholonomic constraints that impose limitations on the dynamically admissible trajectories that a given structure may undergo. The optimality equations for the problem with a quadratic performance index are derived in a compact form involving time derivatives of all modal variables and Lagrange multipliers, which are required to ensure that the constraints are satisfied. These equations are solved by applying symbolic differential operators. The procedure employs standard finite element and symbolic mathematical software to render the optimal actuation forces and trajectories of all controlled modes (or any selected degrees of freedom) to attenuate vibrations. For illustration the method, referred to as the constrained modal space optimal control, is used to synthesize the active controls and predict response of a mast structure.