Optimal control problems for linear dynamic systems with quadratic performance index are solved using the beam analogy. The governing equations for the optimal maneuver are derived in the form of coupled fourth order differential equations in the time domain. These equations are uncoupled using modal variables. Next, each independent equation is made analogous to the corresponding problem of a beam on an elastic foundation. The beam problem in the spatial domain is solved using standard FEM software. Finally the FEM results are transferred back to the time domain where they represent the optimal trajectories and controls for the dynamic system.
SUMMARYThe governing equations of the problem of optimal vibration control of continuous linear structures are derived in the form of a set of fourth-order ordinary di erential equations in the time domain. The equations decouple in the modal space and become suitable for handling by the FEM technique with the time domain subdivided into 'ÿnite time' elements of class C 1 . It is demonstrated that the standard beam element with cubic Hermitian interpolation functions, routinely used in a static analysis of beams, can conveniently be substituted for the required 'ÿnite time' element.
SUMMARYDynamic problems of active optimal vibration control of elastic structures are solved by applying 'static' beam elements from standard structural FEM software. The approach uses an analogy between the optimality equations for the vibration control problem and the equations for the static bending of certain beams. The computer implementation of the analogy, which was presented in Part I of the paper, is discussed here in detail.
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