The analytical construction of displacement functions for plate elements subject to bending was discussed at some length in ref. 1. Attention was thereby drawn to both triangular and quadrilateral elements available in the ASKA system, which satisfy either all kinematic compatibility conditions or are deficient in the continuity of the gradient normal to the edge. The most rudimentary elements of the triangular set consist of the so-called fully compatible TRIB 3C and the TRIB 3, which satisfies the slope condition only at the vertices. The TRIB 3C corresponds in principle, but not in detail, to the element evolved by Bazeley et al but allows, however, also for linear taper.

It was pointed out in ref. 1 that various interpolation techniques could be used to develop elements (for the matrix displacement method) of increasingly more complex characteristics both as far as geometry and specification of strain are concerned. A number of authors have examined this problem in detail and attention is drawn to the excellent work of Bogner, Fox and Schmit on plate elements, and to the elegant paper by Pestel. Over a number of years the ISD at the University of Stuttgart has investigated the application of interpolation functions and has developed within the ASKA language a number of corresponding elements.

Some approaches are indicated for determining, and if necessary raising, the potential utilisation of training equipment, the objectives being:

—to reduce costs;

—to avoid or defer new capital investments;

—to secure more flexible rostering through increased

availability.

The problem is approached from a systems standpoint.

The investigation has been mainly confined to Link equipment, but the solution can be utilised for all auxiliary equipment, e.g. radar, where failures occur during or outside the use phase. Visual aids and pure flight aspects such as motion are not taken into consideration.

The attitude dynamics of a spinning axi-symmetric satellite, free to librate in roll, is studied using the WKBJ method. Applicability of the method to the linearised equation of motion is first verified. To facilitate the evaluation of the WKBJ solution and to make it more amenable to the phase space analysis, second order terms in eccentricity are neglected. The form of the solution suggests the presence of phase and amplitude modulation effects for non-circular orbits. The comparison between the WKBJ analysis and ‘exact’ numerical solutions showed that the approximate analysis predicts the period and amplitude of motion with an accuracy adequate for the preliminary design purposes.

Attitude dynamics of a spinning, axi-symmetric, rigid body undergoing central force motion has received considerable attention in recent times. Thomson et al studied the problem where the satellite was restricted to follow a circular trajectory. This stipulation reduced the system to an autonomous form which was then treated by linearised or Liapounov type of analysis. For the satellite in an elliptic orbit, Kane and Barba presented numerical solution to the linearised equations of motion. Recently Wallace and Meirovitch investigated the same problem by performing asymptotic analysis on linear and low order non-linear systems.

The problem of a two-dimensional pressure field moving over a free surface is analysed by dynamic modelling techniques. Using a simple second order linear differential equation, it is possible to obtain all the classical theory rersults for a uniform field in deep water. Squire’s results for the non-uniform pressure field of a planing wedge are also duplicated. Further results, not hitherto obtained with classical theory, may explain some anomalies associated with the classical theory. The method has the advantage that results for a large number of “pressure field distributions” have already been tabulated by workers studying the response of simple lumped parameter mechanical dynamic systems.

A conclusion of some practical importance in applied hydrodynamics is that the wave drag of a pressure field varies with its distribution. The more non-uniform a pressure field the greater its wave drag. The theory also gives insight into the occurrence of “humps and hollows” in model resistance curves, and gives some insight as to why a minor change in form may cause a major change in wave resistance.

Recently there has been renewed interest in the concept of an all-wing aircraft as a means of producing cheap air transport over relatively short distances. It is natural that with the large amount of information on slender wings now available an all-wing aircraft based on a sharp-edged slender planform should be considered for this role. One of the difficulties immediately faced in developing this concept is that since the aircraft must carry a large number of passengers it is necessary that as much of the wing area as possible should be deep enough to provide for a large passenger cabin. Thus the wing will be very thick over a large part of its area. If this condition is not met, then the aircraft has too much wing area and hence too high a structure weight. Typically one may think of an aircraft with a delta wing of aspect ratio 2 and with a wing thickness of from 15% to 20% of the root chord over as much of the wing area as possible. At first sight thickness of this order eliminates the main advantage of slender wings since the effect of thickness is usually to reduce the strength of the leading-edge vortices and hence the non-linear lift. Thus the incidence for a given lift is increased above that for a thin wing. This in turn means that the lift-to-drag ratio may be smaller.

Following the appearance of my note on this subject in the July 1967 JOURNAL, Professor Hemp kindly pointed out a mistake in one of the basic assumptions. This has led to a recalculation of the results, and the correct results are given below.

a width of beam

b lower bound on height of beam

d upper bound on height of beam

E Modulus of elasticity

h (x) height of beam at position x

H the Pontryagin Hamiltonian

i second moment of area=ah3/12

i length of beam

M (x) bending moment =

rl, r2non-dimensional length parameters

p density al lengthparam

t non-dimensional length parameter=x/l

u non-dimensional length parameter=h/l

x horizontal distance from base of beam

y (x) vertical displacement of the centre-line of beam

Y error function in hill climbing sequence

Vectors

f phase space velocity vector =dX/dt

P adjoint vector

X non-dimensional state vector