Active optimal attenuation of elastic vibration is analyzed by the finite element (FE) technique. A quadratic performance index is adopted to obtain optimality conditions for the problem using Pontryagin's Principle. The FE approach is applied directly to the optimality conditions that are transformed to a set of 4th order ordinary differential equations with specified initial and final conditions in the time domain. It is shown that under some assumptions, the variables in the set de-couple in the modal space, and the optimality condition for each modal variable becomes analogous to a certain static problem of bending of beams. Such an analogy, referred to as the beam analogy, allows for modeling the problems of optimal vibration control by a set of static beams. In turn, the problem of static beams can be handled efficiently in the spatial domain with the help of the standard beam element with the Hermite approximation functions. A simple example illustrates the analogy.
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