Optimal vibration quenching for an Euler-Bernoulli beam

The quenching of the vibration of an Euler-Bernoulli beam under tension wit h general linear homogeneous boundary conditions is studied using a distrib uted control. A method for determining the control that quenches a finite n umber of modes is given and it is shown that the method can be extended the oretically 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 vibr ation 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 det ermining how many modes are sufficient to be quenched if the residual posit ion and velocity of the beam are both to remain within a restricted band af ter the control is removed. Numerical results are given in graphical form. (C) 1999 Academic Press.