ON THE STABILITY OF THE DAMPED HILL EQUATION WITH ARBITRARY, BOUNDED PARAMETRIC-EXCITATION

The minimum damping for asymptotic stability is predicted for Hill's e quation with any bounded parametric excitation. It is shown that the r esponse of Hill's equation with bounded parametric excitation is expon entially bounded. The parametric excitation maximizing the bounding ex ponent is identified by time optimal control theory. This maximal boun ding exponent is balanced by viscous damping to ensure asymptotic stab ility. The minimum damping ratio is calculated as a function of the ex citation bound. A closed form, more conservative estimate of the minim um damping ratio is also predicted. Thus, if the general (e.g., unknow n, aperiodic, or random) parametric excitation of Hill's equation is b ounded, a simple, conservative estimate of the damping required for as ymptotic stability is given in this paper.