On the criteria for the stability of small motions

Abstract

When a dynamical system receives a small disturbance from a state of rest or steady motion, the ensuing small motion is governed by a system of linear differential equations. In order to determine the stability, the conventional procedure is to examine the signs of certain "test functions," which can be constructed in succession from the coefficients of the determinantal equation by Routh's well-known rules (see Routh's "Rigid Dynamics," vol. 2, 6th ed., p. 228). However, the series of test functions for a determinantal equation of general degree are not stated by Routh in an explicit form; and the expressions would, in fact, be exceedingly cumbersome. An alternative is to use for the stability criteria the signs of certain "test determinants." This method, which is very convenient in practice, is not described in works on dynamics known to the writers, and may be novel. The present paper contains a brief account of these determinants and of certain other simple forms of test function. The stability of a system is usually dependent upon so many factors that the exact influence of individual factors may be extremely difficult to trace in a purely algebraic discussion of the test functions; but such obscurities can often be avoided by a graphical representation of the criteria. A suitable graphical treatment for problems of a certain wide class will be described. For a detailed illustration of the application of the method to the stability of aeroplane wings, the reader is referred to R. and M. 1155.