Formulation of “Stability Equations” and Derivation of New Constructions with Complete Systems of Design-formulae for Variable-speed Control Mechanisms

Abstract

Requirements for stability are formulated mathematically and, through the “transformatory operations of mathematics”, yield a series of “stability equations” of ascending order which are generally applicable, for example to control mechanisms, electronics†, nuclear physics, etc. From these stability equations, the equation of the stable characteristic curve of a governor, and the differential equations of the oscillations of a governor-engine system, are derived. It emerges that the first part of the new oscillatory equation is identical with the whole of the differential equation in the literature to date (unchanged since Maxwell 1868)‡, while the important second part, which consists of terms of the same order of magnitude as the first part and which is the only one containing the equation of the stable characteristic curve, is lacking in literature. The stability equations classify all possible constructions of variable-speed governor according to “order of stability”, which signifies important operating properties. This classification accounts for the known shortcomings of conventional types. The stability equations, combined with the mathematical formulation of practical requirements (speed-adjustment with only one actuating motion, etc.), lead to new basic types of variable-speed governor, with complete systems of design equations. In addition to determining all unknown dimensions, this set of equations is important because it derives constructions of which the complexity increases with order of stability and, furthermore, a simple construction which provides any required high order of stability with the minimum number of adjustable components.