On the Stability of Accelerated Motion: Some Thoughts on Linear Differential Equations with Variable Coefficients

Summary:

In studies of the stability of aeronautical systems, equations of motion are derived which have coefficients dependent on flight speed. Conventional practice treats the speed as constant, when a set of linear differential equations with constant coefficients results. Actually, since the speed varies during flight, it may be regarded as a prescribed function of time; the set of linear differential equations then has variable coefficients.

The treatment of the problem of stability then becomes much more complex in this case. A simple example is given to show that a system which is stable at any constant speed can become unstable during deceleration; the ordinary constant-speed criteria are, strictly, therefore inadequate. Some approaches to the discussion of stability during acceleration are suggested; a solution is given of the single second-order equation which enables the amplitude of oscillation of the solution to be studied. Inverse methods of approach are suggested, both for single and sets of equations, in which particular forms of acceleration corresponding to prescribed solutions are derived; and some tentative conclusions are drawn. As would be expected, the effects of acceleration depend on a dimensionless “acceleration number.”