Notes on the analysis of stability in accelerated motion

Abstract

Linear differential equations whose coefficients are functions of the independent variable are now assuming importance in aeronautics in the discussion of the motion following a small disturbance in a specified accelerated motion. In such problems the undisturbed state is often a transition motion in a limited time interval between two steady motions and we are concerned to see that the disturbed motion does not exceed tolerable bounds in a limited time. The extension of classical stability theory to such problems involves some logical difficulties and very great mathematical ones, since such equations are seldom soluble algebraically. For these reasons an indirect attack oll the problem is made here by seeking to establish upper bounds to the solution of second-order equations, which are those most commonly occurring. The subject is introduced by a study of the equation x.. + b(t)x. + c(t)x = 0. The theory is then applied to a simple problem of pitching motion in an airstream of varying velocity. Finally a system of two second-order equations involving two variables x and y is discussed from this angle. This system is not tractable in its most general form, but the special cases that yield to treatment are those which have occurred in some recent problems. This analysis should be useful in the examination of any problem to which it can be applied; the exploration of its range has hardly begun. It is, of course, open to the objection that the gap between the bound and the solution can in general only be found by numerical integration. Some surveys by numerical integration to compare with the bound analysis will be the quickest way of assessing this method as a tool for general use.