The Solution of Lagrangian Frequency Equations with Complex Coefficients or Roots by the Escalator Method

Abstract

THE orthodox solution of Lagrangian frequency equations involves the expansion into polynomial form of the characteristic determinantal equation in the latent roots, but this method becomes exceedingly laborious if a large number of frequencies and their associated modes are required accurately for any system of equations of high order, say above the sixth. We define a system of Lagrangian frequency equations to be of the nth order if it consists of n equations for n homogeneous unknowns, which we call modes. A useful contribution to the problem was made by the iteration solution of Duncan and Collar, which is especially valuable when only the highest one or two latent roots are required. But when an aircraft propeller vibration problem required the first seven frequencies and their associated modes for a 12th‐order equation whose coefficients involved a variable pitch angle, the labour of calculation by this method appeared at that time (1941) to be prohibitive. The ‘Escalator’ method was therefore devised jointly by the author and Captain J. Morris of the Royal Aircraft Establishment as an alternative. In the propeller problem all the latent roots involved were necessarily real. Dr L. Fox, using relaxation methods, has recently solved a similar problem in a remarkably short time. Unfortunately, relaxation methods cannot easily be extended to the case of complex latent roots, which can occur in connexion with flutter, radio circuits and other problems. In this paper it is shown how the Escalator method can be adapted without essential change to cases in which complex quantities occur.