The two-dimensional stiffened plate problem is completely governed by the fundamental differential equation

(22)

where k1 and k2 are two non-dimensional structural constants depending on the relative proportions of the reinforcing members.

The complete solution is

(25)

This solution renders all solutions for the isotropic plate available for the stiffened plate also. The choice between the several types of solution available is governed by the same considerations as for the isotropic or unstiffened plate.

This paper describes the construction of a water channel in the University of Bristol for the investigation of the analogy between the two-dimensional flow of a gas and that of shallow water with a free surface. Both continuous and discontinuous flow were examined, with a view to determining the limitations of the analogy.

Continuous “ shooting “ water flow was found to be reasonably analogous with supersonic isentropic gas flow, a static depth of about half an inch appearing to be satisfactory with this particular channel. No independent check was made of the agreement, or otherwise, between “streaming” water flow and subsonic gas flow, since the method of checking used was the measurement of the angle of the waves formed on the water surface, and such waves exist only in “ shooting” flow.

The effect of yawing a wing in high-speed flow is to delay the onset of shock waves and to increase the critical Mach number. This is because shock waves can only develop along the isobars (running parallel to the edges of the wing, which is assumed untapered and infinitely long), and therefore only the velocity component normal to the edges is significant. To some extent the same is true for a finite sheared or swept-back wing, though the problem is made more complicated by the various additional factors, such as finite aspect ratio (tip effect), plan form (e.g. taper effect) and, what is perhaps the most important, the central kink effect. The experimental technique is extremely cumbersome because of the many shape parameters involved (see Figs. 1, 2, 3), and since analytical solutions present fundamental difficulties and pitfalls, designers tend to favour rule of thumb methods. The present paper contains a general review of the theoretical work done at The Royal Aircraft Establishment, which was limited to the case of zero incidence. This limitation is not a severe one, as flight at high speeds often implies low incidence. Incidences which are not negligible involve the solution of another fundamental problem (that of the lift distribution).

The beam on elastic supports is often considered as a beam on a continuous elastic medium. This method can be applied only to a large number of equally spaced supports having equal rigidities. The method proposed by the author, giving the solution in the form of rapidly converging series, is not bound by this restriction, and when applied to the problem of beams supported by cross-girders reduces it to a system of linear equations, their number being equal to the number of cross-girders, while the existing approximate method leads to the same number of simultaneous differential equations.

This paper has been compiled because of the applicability of the treatment of the stability of the motion of the common spinning top to the problem of the whirling of rotating shafts, carrying loads of appreciable moments of inertia. This problem is assuming renewed interest and importance, especially in the drives of contra-propeller systems and the more recent high speed prime movers.

The method of analysis adopted by Cameron has been extended to develop more general equations for the correction of level speed and rate of climb for variation of air temperature and aircraft weight, applicable to aircraft with reciprocating engines of any type driving constant-speed propellers, operating at speeds sufficiently high for compressibility effects on drag and propeller efficiency to be appreciable. Allowance is made in the equations for ram effect on engine power.

The practical application of the equations is discussed, and illustrated by a worked example. The influence of errors in the estimation of the parameters expressing the variation of engine power and propeller efficiency on the corrections, in this particular case, is discussed.

A method of design is given for wind tunnel contractions for two-dimensional flow and for flow with axial symmetry. The two-dimensional designs are based on a boundary chosen in the hodograph plane for which the flow is found by the method of images. The three-dimensional method uses the velocity potential and the stream function of the two-dimensional flow as independent variables and the equation for the three-dimensional stream function is solved approximately. The accuracy of the approximate method is checked by comparison with a solution obtained by Southwell's relaxation method.

In both the two and the three-dimensional designs the curved wall is of finite length with parallel sections upstream and downstream. The effects of the parallel parts of the channel on the rise of pressure near the wall at the start of the contraction and on the velocity distribution across the working section can therefore be estimated.

The statement that with modern aircraft and jet propulsion engines it is essential to match the engine and aircraft has become a platitude, without the reasons being quite so well appreciated. It is well known that should the aircraft be operating away from its “ design point,” involving throttling back of the engine, then the loss of efficiency with either jet or turbine airscrew engines is liable to be serious, and to have a large effect on the range. There is also a firm conviction that very high altitude is necessary in order to give a jet aircraft any range worth mentioning.

It seems worth while to examine in a simple way on what qualities of the engine the range depends. The object of this simplified attack on the problem is two-fold; first of all it enables the factors to be readily appreciated, and second it provides rapid methods of comparing the qualities of two different engine types for range.

It is shown that the rotational stiffness of a crossed flexure pivot varies considerably when subjected to an applied force. The type of variation can be radically changed simply by moving the point at which the strips cross. The relation between torque and rotation for a given applied force is not exactly linear and the extent of the non-linearity is determined by taking into account the small movements of the centre of rotation of the pivot. Finally, for design purposes, an analysis of the maximum stresses in the strips is given.

Crossed flexure pivots, of the type shown in Fig. 1, are extensively used in aeronautical research equipment to replace knife edges or ball bearings. Perhaps the most frequent use is for the fulcrum of the balance arm in a wind tunnel balance. Here, a null-point method of reading is almost invariably used, so that the prime considerations influencing the design of a flexure pivot for such a purpose are (a) that the applied loads should not cause instability of the pivot, and (b) that the rotational stiffness should be small enough to give adequate sensitivity.

Following the original work of Munk, Jones, and Ward for steady flow, a solution is given for unsteady, supersonic flow over very slender bodies of revolution and wings. The results are subject to the restriction (M2- l)δ2 loge [(M2- 1)½ δ]«1 where δ is the slenderness ratio, M is the Mach number and, in addition, to the usual restrictions imposed by linearisation. As examples, the lifts and pitching moments on flat wings and bodies of revolution executing pitching motion and the damping moment on a rolling wing are calculated.

It is shown that the order of approximation is consistent with the limitations already imposed by linearisation, at least in supersonic flow, where no Kutta condition is required.

When a uniform, two-dimensional supersonic flow expands suddenly round a corner in a wall it forms a pattern known as a Prandtl-Meyer expansion or centred simple wave. If the flow is two-dimensional but not initially uniform, or if it is axially-symmetrical, the expansion is still centred, but is not a simple wave. An approximate solution is given in this paper for the isentropic, irrotational, steady two-dimensional or axially-symmetrical flow of a perfect gas in the neighbourhood of the centre of such an expansion. The solution is designed to replace the conventional method of characteristics in such a region.

The main application is to a jet issuing from a nozzle that discharges into a container with a pressure lower than that in the nozzle; in particular, a formula is derived for the initial curvature, at the lip of the nozzle, of the boundary of the jet. The solution also applies to the flow near an edge in a boundary wall, and a formula is derived for the velocity gradient on the wall immediately downstream of the edge.

Performance measurements for turbo-jet and turbo-prop aircraft are most conveniently reduced to standard conditions by the use of graphical methods. The case of rate of climb presents difficulties if readings are taken in the normal way during continuous climbs, because of the variation of weight, temperature, pressure and optimum climbing speed with height.

Since rate of climb is proportional to the product of speed and acceleration in level flight, these quantities could be measured and used to determine the rate of climb at any height.

The advantages of the method are listed and the technique required of the pilot examined. An automatic observer, including a longitudinal accelerometer, reading, in some cases, to within 0.001g, is necessary.

It is shown to be a consequence of kinematics and statics that the aerodynamic derivative expressing the rate of change of hinge moment of a plain flap with the angular setting of the flap varies parabolically with the chordwise position of the hinge. The derivative expressing the rate of change of hinge moment with incidence of the main surface varies linearly with the hinge position. Similar conclusions hold for the derivatives of the hinge moment with respect to the corresponding angular velocities.

A linearised theory for the subsonic, lifting surface problem is formulated in terms of Fourier integral solutions to Laplace’s equation. The symmetric and anti-symmetric problems of the first kind are solved explicitly, while the problems of the second kind depend on the solution of dual integral equations. The antisymmetric problem of the second kind is cast in a variational form, from which certain well-known theorems may be deduced.

The paper is concerned with the two-dimensional, steady, irrotational, isentropic flow of a perfect gas in a wind tunnel nozzle which is found to produce a flow in the test rhombus deviating slightly from the desired uniform flow.

The minimum corrections are derived that must be applied to the liners in order to produce a uniform flow in the test rhombus. If the uncorrected nozzle produces a flow of uniform direction, measurement of the pressure on the axis, in the test rhombus, suffices to determine these corrections (Section 5). If not, further pressure measurements are required (Section 6). A simple test is indicated for checking whether the flow stream direction is uniform (Section 6).

The method cannot be used to correct for deviations from a two-dimensional flow.

As is well known, standard lifting-line theory cannot be applied to a wing with sweepback in order to determine the amount and distribution of the lift. Various methods, all fairly laborious, are available, and while they give results which agree fairly well with experiment, they do not show in detail the source of the various differences due to sweep. It is with the object of separating the several effects involved, which can be regarded as distinct for practical purposes, that the present method has been developed. Since one of the items, finite-chord loss, is shared with the straight wing, although not included in simple lifting-line theory, it is calculated first for the straight wing. The wing is represented by its median or camber surface and is assumed nearly flat.

The present note deals with the peculiarities of the lift distribution on swept wings. The influence of the angle of sweep is expected to be greatest in its effect on the chordwise lift distribution at the centre section of the wing. It is assumed that the essential characteristics of this chordwise lift distribution are predominantly determined by the special configuration of the bound vortices and not so much by the trailing vortices. In calculating the downwash, induced by a continuous chordwise distribution of swept vortex lines of infinite span, it is found that the downwash at the centre line is to be expected to differ from its two-dimensional form mainly by a term proportional to the local vortex strength. Solutions of this simplified downwash equation have been obtained. They differ from the ordinary fiat-plate distribution only in the exponent which instead of becomes .It is suggested that this type of function is more appropriate for the centre-section of swept wings and that it may replace the ordinary flat-plate distribution in all those cases where the lifting surface is swept.

In a theoretical flutter analysis, the real zeros x˂0,y˃0, of a stability determinant Δ(√x, y) which is a polynomial in y are required, and can readily be evaluated once the polynomial expansions are known for assigned values of x. When the air loads are estimated on the basis of classical derivative theory, Δ becomes a bivariate polynomial in √x and y of a special type. Direct expansion of the stability determinant becomes too involved when the number of degrees of freedom is large and indirect methods of expansion are needed.

A method of bivariate expansion which uses a framework of lines subject to certain restrictions is examined in Part I, and is applied to a quaternary wing-flutter problem. The analysis shows that frameworks previously proposed, having all intersections outside the flutter quadrant (x˂0, y˃0), demand very high computational accuracy. A new framework is suggested having all intersections inside the flutter quadrant. The improvement in computational accuracy is considerable.

In the previous linearised theories of steady supersonic flow past a slender oscillating aerofoil, the shock line at the nose has been assumed to be a straight, stationary one, so that no vorticity is produced on passing through it. In the present treatment, the possibility that the shock may be both curved and moving is taken into account, but the result is exactly the same as that obtained by the previous treatments (e.g. Temple and Jahn).