On fundamental sets of solutions of the equations of lateral motion, and the rapid calculation of general solutions

Abstract

The lateral motion of a symmetrical aeroplane slightly disturbed from steady flight is determined, to the first order of small quantities, by the solution of a system of six simultaneous linear differential equations with constant coefficients, in which the inhomogeneous terms, representing control forces or the effects of gusts, may be arbitrary functions of time. In virtue of the general properties of such equations, as is well known, their most general solution can always be written down in a form involving definite integrals. Calculations of such theoretical expressions can be very tedious, and it is now shown that the most general solution can be much more simply obtained, by processes of addition, multiplication, and integration, from a set of three fundamental solutions. A large number of such sets of fundamental solutions has already been obtained by means of the differential analyser, and the application to these of the methods of this report will make possible a large range of more special response calculations, some of which may well develop into important matters of routine. After an introductory statement of the equations of motion, the three fundamental solutions are defined in sect. 3.1, with four further solutions which are conveniently regarded as fundamental, though they can be derived from the original three. Relations between these seven solutions are given in sects. 3.2 to 3.6. Sect. 4 is concerned with the derivation of other solutions corresponding to constant or piecewise constant disturbances, and generalisation to disturbances given as any functions of time is made in sect. 5. A few particular examples of the technique developed are given in sect. 6, the fundamental solutions used being chosen from the differential analyser results mentioned above. A brief account of the scope of these is given in an Appendix, which includes in tabular form an index to the complete series of 1188 figures in which the results are contained.