Numerical Damping of Forced Oscillations of an Elastic Beams

The beam oscillations are modeled by the fourth-order hyperbolic partial differential equation. The minimized functional is the energy integral of an oscillating beam. Control is implemented via certain function appearing in the right side of the equation. It was shown that the solution of the problem exists for any given damping time, but with decreasing this time, finding the optimal control becomes more complicated. In this work, numerical damping of beam oscillations is implemented via several fixed point actuators. Computational algorithms have been developed on the basis of the matrix sweep method and the second order Marquardt minimization method. To find a good initial approximation empirical functions with a smaller number of variables are used. Examples of damping the oscillations via a different number of actuators are given. It is shown that the amplitude of the oscillations of any control functions increases with the reduction of the given damping time. Examples of damping the oscillations in the presence of constraints on control functions are given; in this case, the minimum damping time exists. The damping of oscillations is considered also in the case when different combinations of actuators are switched on at different time intervals of oscillation damping.