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Title: Mathematical Models for Finite Controllers for a Class of Bounded Acoustic and Structural Dynamic Systems

By: Panza, M.J.

Date: 2004

Publication: Proceedings of ACTIVE 2004, Williamsburg, VA

Abstract:
A class of acoustic and vibratory systems having partially bounded regions causing multiple reflections is considered. Due to the infinite number of system modes associated with these regions, an infinite number of modal dimensions are required to describe the system dynamics. Active control to supply system damping and reduce the peaks in the transfer function over a broad range of frequencies generally requires many actuators to control the many system modes. A closed form representation of the system dynamics may allow for a finite model-based controller to be applied using one actuator. The acoustic space between two parallel reflecting planes and a simply supported beam are used to represent the class. The approach applies the Euler-MacLaurin sum formula to the infinite series Green function description of the space to obtain an approximate closed form expression. The concept is demonstrated by using the model with a basic feedforward control scheme to meet a desired broadband output frequency spectrum. The method provides a finite controller in Laplace transfer form. The main contribution is two applications of the Euler-MacLaurin sum formula to go from an infinite dimension Laplace transform modal description of the system Green function to an infinite dimension Laplace transform image description to an approximate finite closed form solution in the time domain. The Laplace transform of the closed form solution for the Green function is applied to the system equations with disturbance and control. Numerical results demonstrate the utility of the models.