This note attempts to obtain an approximate solution for a class of second-order systems represented by

where the well-known procedures, such as perturbation, variation of parameters, the WKBJ analysis, etc, may prove to be inadequate. The approach is illustrated through an example related to the planar librations (satellite oscillations about its mass centre) of gravity gradient cylindrical satellites in circular orbit, subjected to the solar radiation pressure damping (Fig. 1).

Two papers by Dixon described a digital computing technique for the structural optimisation of a cantilever beam. This technique applied Pontryagin's Principle to the problem, and computation was effected using a modified Kutta-Simpson integration technique and the Rosenbrock hill-climbing method. Dixon's second paper, in which he revised his earlier work, cited the hybrid solutions of the authors for comparison with his own results. This comparison of the results obtained from the digital and hybrid methods showed a slight, but significant, difference which indicated the need for further verification.

It has been known for some time that when a cylinder spans a wind tunnel the boundary layers on the walls of the tunnel may have important effects, especially on the pressure in the base region. By making the boundary layers thin by the use of end plates, it may be possible to approach a complete elimination of the effects of these layers on the central part of the span and obtain a flow that is very nearly two-dimensional. Most of the published pressure distributions for circular cylinders were obtained without end plates and this probably explains their inconsistency. Gerrard has described experiments in which the addition of end plates caused a substantial change of base pressure coefficient but he did not attempt to design an optimum end plate to give nearly two-dimensional flow.

In an early investigation of gas flow through a spool valve model Stenning discovered the same bi-stable flow patterns as had been established for oil flow, but reported no measurement of quantitative similarity or of other compressibility effects which would be of importance if such valves were to be used in pneumatic systems. Since then there have been no reports of comparisons between compressible and incompressible flow patterns in control valves, and the present investigation has been carried out to determine the characteristics of compressible flow through a two-dimensional model of a control valve port and to try to derive these characteristics from established compressible flow behaviour.

The mathematical analysis of the induced velocity components of a screw propeller has occupied the attention of fluid dynamicists for many years. Early approaches to the problem concentrated mainly on momentum and blade element theories, neither of which are capable of providing a detailed knowledge of the flow field, except at the propeller plane or the boundary of the fluid. The early vortex theories also suffered from equally significant, but less severe, limitations and such methods were found to be completely unsatisfactory for many applications. In particular, the development of the ducted propeller for the propulsion of supertankers and VTOL aircraft has demanded a highly detailed knowledge of the flow field, especially around the propeller tip.