Optimal Vibration Quenching for an Euler]Bernoulli Beam

The quenching of the vibration of an Euler]Bernoulli beam under tension with general linear homogeneous boundary conditions is studied using a distributed control. A method for determining the control that quenches a finite number of modes is given and it is shown that the method can be extended theoretically to determine a control to quench all modes of the vibration. In general there is more than one control that can be used to quench the same modes. It is shown that of all controls that quench specified modes of vibration at a given time and are square integrable the method described yields the unique control whose mean square is minimum. A method is given for determining how many modes are sufficient to be quenched if the residual position and velocity of the beam are both to remain within a restricted band after the control is removed. Numerical results are given in graphical form.