Numerical damping of oscillations of beams by using multiple point actuators

Methods of damping the oscillations of complex mechanical systems’ elements, such as strings, membranes, beams, plates, have intensively been developing since the 1970s. In particular, the beam oscillations are modeled by the fourth-order partial differential equation, which is Petrovsky-hyperbolic. 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. Previously, it was shown that the solution of the problem exists for any given damping time, but as soon as this time decreases, the optimal control becomes more complicated. To obtain approximate numerical solutions, the so-called point actuators were considered. Control was considered via single point actuator placed at some point of the beam, but in this case it turned out that it is not always possible to dampen it. Therefore, control was also considered via point actuator moving along a small section of the beam. However, the implementation of such an actuator is very difficult. In this work, the 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 of finding the minimum of functions of many variables. To find a good initial approximation, when minimizing the energy integral, empirical functions with a small number of variables are used. This made it possible to significantly reduce the calculation time of the task. The 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. The 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.