Application of optimization methods to vibration control of stretched strings

Abstract

In this paper, non-linear programming techniques are applied to the problem of controlling the vibration pattern of a stretched string. First, the problem of finding the magnitudes of two control forces applied at two points l₁ and l₂ on the string to reduce the energy of vibration over the interval (l₁, l₂) relative to the energy outside the interval (l₁, l₂) is considered. For this problem the relative merits of various methods of non-linear programming are compared. The more complicated problem of finding the positions and magnitudes of two control forces to obtain the desired energy pattern is then solved by using the slack unconstrained minimization technique with the Fletcher-Powell search. In the discussion of the results it is shown that the position of the control force is very important in controlling the energy pattern of the string.