A method is presented for the analysis of a simply-supported rectangular orthotropic plate subjected to a uniformly distributed load by the application of Maclaurin's series. The solutions have practical applications in the design of interconnected beam systems because of the elastic equivalence of the grid framework to an orthotropic plate.

The application of Maclaurin's series to the analysis of beams in bending was first described by Hetényi. Recently Reddy and Gangadharan described a modified approach for beams with stepped variations in cross section. Ballesteros has extended the method to functions of two variables in the analysis of isotropic plates in bending.

In the past few years pilots landing in very heavy rain, or just after it, have found their wheel-brakes ineffective due to the layer of water, and this has been called aquaplaning; the word is preferable to “hydroplaning” which already has a well-established meaning in a deep-water context. The nature of this phenomenon has been speculated upon by NASA since 1959 and by Mr. Gadd of the NPL in 1961 (see R.Ae.S. Journal, March 1963). The latter gives the impression that in Britain we are still speculating, but in fact we know quite a lot about it now; much has been described in an RAE Technical Note last November and this was added to at a Take-off and Landing Symposium at Farnborough in December 1962. These sources are being drawn upon here.

The basis of the procedure described in the paper is the replacement of the stiffeners by an approximately equivalent system of line springs. One of two methods may then be used to determine the natural frequencies. A rectangular plate with edge stiffeners, point-supported at the four corners, is used to demonstrate the application of the Rayleigh-Ritz method. Numerical results obtained are compared with known approximate solutions based on finite difference equations. A Holzer-type iteration is employed in the case of a plate with parallel stiffeners, where the two edges perpendicular to the stiffeners are simply supported, the other two edges having any combination of conditions.

In the art of brake design, many of the difficulties which existed in the past have now been overcome, and this is due largely to the wealth of experience obtained over a wide range of energy absorption rates and heat sink temperatures. Much of this experience has been obtained on brake test machines on which the essential aircraft braking conditions can be reasonably simulated on the complete assembly of wheel, tyre and brake.

For a given plate and loading the optimum supporting ring radius is shown to be approximately 0·677a, there being a slight variation with Poisson’s ratio.

The “edge square” condition, i.e. zero slope at the circumference of the plate, is also investigated. It is satisfied by a ring radius of , a result which is independent of Poisson’s ratio.

Macaulay’s brackets are used in the analysis as an example of their value in circular plate problems.

In most aerodynamic problems the speed of sound, a, is naturally taken as or where p, ρ and T are static values, γ is the specific heat ratio and R is the gas constant. This particular speed, however, is specifically that of plane waves advancing uniformly into an infinite expanse of a homogeneous medium. There are many other situations where the medium is bounded or inhomogeneous, for instance, in which small pressure waves do not propagate at this speed relative to the medium. Phenomena such as choking can then be shown to occur, not when the flow velocity is exactly equal to but when it reaches the new wave speed appropriate to the local circumstances.

Rather more than twenty years ago the writer published a report on the design of wind tunnel fans: a single fan was implied. The report assumed that the pressure increment appropriate to a given axial wind velocity was known, having been either measured (e.g. on a model), or estimated from data such as the power factor of the wind tunnel. A preliminary calculation to determine rotational speed and solidity was followed by a detailed calculation of the appropriate blade angles; the detailed calculation involved the evaluation of the rotational inflow factor (here called a: in Ref. 1 it was denoted a2). Now most of the quantities involved are functions, implicit or explicit, of the rotational inflow factor, and it is not possible to evaluate it in closed form: some method such as cross-plotting for ranges of a at each radius, or successive approximation by a convergent process, is required. Ref. 1 advocated the latter method.

Investigations by one of the authors in connection with the design of a fan for a blower type of wind tunnel showed that regular and repeatable dust patterns occurred on the blades of a one-quarter scale model fan of 18 inches diameter. Dust was deposited on the fan blades along the leading-edge and on the suction surface over an area thought to be the turbulent region of the boundary layer. The introduction of isolated protuberances on the dust free area of a blade gave rise to turbulence wedges in which dust was also deposited and this was interpreted as confirmation of the coincidence of the dust deposits with regions of turbulent boundary-layer flow. These deposits showed the existence of a considerable extent of laminar flow on the suction surface of each blade close to the root, a region where high lift coefficients would be expected with associated adverse pressure gradients. Two-dimensional wind tunnel experiments were made to confirm the interpretation of the observed dust patterns by comparison with the smoke filament and volatile liquid methods of flow visualisation and these are reported in Reference 2.

This investigation has been concerned with heat transfer between a jet and the inside surface of a circular cylinder whose axis was normal to the jet axis. Such a jet system could be used to direct cold air on to the inside surface of a sheet metal nozzle guide vane in a gas turbine engine. As will be seen later high rates of heat transfer can be obtained in the vicinity of the point of impingement, so that this method of cooling could be particularly valuable for reducing the temperature of a “hot spot.”

The following are various aircraft requirements which, it can be said, a pilot considers in the course of his employment in agricultural aviation.

• 1. Physical comfort.

• 2. Accident protection.

• 3. Operating, performance and flying requirements.

• 4. Work capacity and operational efficiency.

The free flexural vibrations of rectangular plates with various boundary conditions have been considered by Warburton. The natural frequencies were calculated by the Rayleigh method, the mode assumed being the product of the characteristic beam functions for the given boundary conditions. Comparison with experimental results shows that the method gives reasonably good approximations. The present note describes a method of obtaining the approximately equivalent characteristic beam functions to enable Warburton's method to be extended to plates having one or more stiffeners parallel to an edge. As a numerical example expressions for the frequencies are derived for a plate, simply supported along two opposite edges, and having a central stiffener parallel to the other two free edges. The results are compared with those given in a recent note by Kirk, who solved the same problem by the Rayleigh-Ritz method, using a mode with one arbitrary parameter. In the case of the fundamental frequency of the unstiffened plate, the characteristic beam function in a direction perpendicular to the free edges is simply a constant, and the solution is less accurate than that given by the Rayleigh-Ritz method. However, numerical analysis of a square plate shows that above a certain stiffener depth the characteristic beam function method is more accurate than the Rayleigh-Ritz method. The two methods are also compared for the 2/2 mode.

References 1 and 2 discuss the shock position in over-expanded nozzles, and in particular the transition from nozzle flow characterised by a normal shock in the nozzle to nozzle flow characterised by oblique shocks and separation from the wall. As is well known, the shock position and pressure distribution for unseparated overexpanded flow can be adequately explained using one-dimensional fluid mechanics. Ref. 2, while suggesting a criterion for transition to separated flow, maintains that the separation point is not predictable. The criterion suggested by ref. 2 is that whenever the pressure ratio p2/p0, associated with expansion to the nozzle exit plane followed by normal shock compression at the exit Mach number exceeds the nozzle pressure ratio pb/p0, separated flow will occur. Based on this criterion and the double valuedness of the p2/p0 locus, it is argued that at a nozzle pressure ratio of 0·624, a continuous increase of nozzle exit to throat area ratio will provide for transition from unseparated normal shock flow to separated flow and back to unseparated normal shock flow.

The friction force in many physical systems has been found to depend on a power of the velocity, the magnitude of which varies according to the damping mechanism. This note employs an approximate theory applicable to small damping to examine the free vibration of a single degree of freedom system in which the damping force is proportional to an unspecified power of the velocity.

Correlation with experimental data on the internal damping of solid materials shows that this phenomenon also tends to obey a power law although discrepancies exist between the expected and observed behaviours. These arise from the interpretation of the damping coefficient and from the frequency dependence of the dissipated energy.

The loss of torsional or flexural stiffness due to thermal stresses in a thin solid wing is now well known and has been considered by a number of authors, e.g. Dryden and Duberg, Vosteen and Fuller, Bisplinghoff, Hoff, Budiansky and Mayers, Argyris, and the present author. Exact analyses are available only for the “one-dimensional” case of a wing of infinite aspect ratio with arbitrary chordwise distribution of wing thickness and temperature. The loss of stiffness in a similar wing of finite aspect ratio is not as great and, in particular, the proportional reduction in flexural stiffness in a low aspect ratio wing will tend to be less than the proportional reduction in torsional stiffness.

If it is assumed that the effect of radial flow can be neglected, then the vortices which are shed from the trailing edge of a propeller blade can be said to lie approximately on helical lines. These vortices form a vortex sheet which for practical purposes is assumed to extend unaltered downstream to infinity. If the propeller blade is approximated by a lifting-line of bound vorticity, then the velocities at the blade induced by the trailing vortex system can be derived from the simple flow conditions in the hypothetical far wake where the vortex sheets extend equally to infinity in both directions. Results are already well-known for the case where the vortex sheets exist in free space. It is the purpose here to examine the effect of adjacent boundaries, such as the hub and the duct wall of a ducted propeller. The results also relate to a test propeller in a tunnel.

In reference 1 a general analysis was presented for the calculation of thermal stresses and deflections due to temperature gradients in long rectangular plates having lengthwise edges supported elastically. The edge support was idealised as equivalent rotational and extensional stiffnesses depending on the elastic characteristics of the surrounding structure. The special case of infinite extensional stiffness and freely hinged edges (see Fig. 1) was discussed in greater detail and a number of graphs were included for the calculation of edge reactions in terms of the thermal force and thermal moment parameters. Since these graphs used an excessive number of non-dimensional parameters their practical application is unwieldy; four non-dimensional parameters were used instead of three parameters which would be sufficient to describe the required functional relationship. This note shows an alternative presentation of the results which can be used more conveniently in any design calculations.

On reading the paper by D. J. Bell in the February edition of the Journal, I noticed that there was no mention of an adaptation of the energy height approach to solutions of maximum range trajectories. I was surprised to find no documentation concerned with this approach and would like to indicate this technique which I have used several times.