The eigenvalue equations for symmetric vibration of a complete aircraft are derived in a very general form. The “lumped mass” approximation to the continuous mass distribution is used and sub-matrices are associated with properties of relatively simple branches of the system. The final eigenvalue equations are expressed in terms of these sub-matrices, so that in a numerical application the physical system, as such, is considered only in relation to the properties of the simple branches. It is assumed initially that the aircraft wing and tail have flexural axes of the conventional type, but it is shown in the Appendix that under certain conditions the treatment can be generalised to cover swept and cranked wing aircraft.

An investigation is made into the possibility of the completely automatic longitudinal control of an aircraft in the deck landing approach. The method presupposes the existence of a radio beam system transmitted from the carrier and a study is made of the motion of the aircraft down this beam under automatic control.

The longitudinal motion alone of the aircraft and carrier is considered. The motion of the carrier is assumed to consist of a steady forward velocity modified by a non-pitching sinusoidal oscillation in the vertical plane.

The most successful method of approach control proposed is based on the use of a single beam emanating from a source at a fixed distance above the carrier deck and fixed in direction. The preliminary pilot-controlled approach is assumed to be down the mean position of the beam, and the auto-control requires, over and above certain ideal variations of elevator angle and thrust, additional types of control of the following forms. On the elevator, displacement pitch and displacement track error control; and on the thrust, displacement track error control.

Information is also forthcoming regarding the unsuitability of certain other types of control which at first sight seem promising.

It is now more than 25 years since Coker and Filon first developed photoelasticity as a practicable method of exploring the stress distribution in certain types of engineering components. All engineers must be familiar with the striking photographs of fringe patterns obtained by their method in stressed plates of various shapes viewed in a polariscope. In those early days observations were made on Xylonite models and the method was only applicable to components in which stress distribution was of a two-dimensional type. Moreover, with the apparatus then available investigations were somewhat difficult and many engineers at that time were very distrustful of the results obtained from what was, to them, a little understood laboratory technique.

A method is described whereby, at any point in an infinite parallel annulus, the approximate axial velocity due to a single row of high aspect ratio blades may be calculated from a knowledge of the conditions of flow adjacent to the blades. The method is based on the assumption of a simplified expression for the radial velocity, being the product of an unknown function of the radius and an exponential term independent of the radius containing an undetermined constant; the function and the undetermined constant are calculated by reference to the conditions of flow in the plane of the row considered. The flow due to any number of rows is then obtained by summing the radial velocity fields due to each row and obtaining the axial velocities by integration of the equation of continuity.

The solution of the problem with infinitely many rows is shown to have a simple form by virtue of the fact that the flow (provided that the velocities remain finite) settles down to a pattern which is periodic by one stage pitch.

It is assumed that pure anti-symmetric vibrations of an aircraft can exist, involving fuselage torsion but excluding fuselage bending. With this assumption, which in most cases is a reasonable approximation, the eigenvalue equations for anti-symmetric vibrations of a complete aircraft are derived in a very general form. The “lumped mass” approximation to the continuous mass distribution is used and sub-matrices are associated with properties of relatively simple branches of the system. The final eigenvalue equations are expressed in terms of these sub-matrices so that in a numerical application the physical system as such is considered only in relation to properties of the simple branches. It is assumed initially that the aircraft wing and tail have flexural axes of the conventional type, but the treatment is generalised to cover swept and cranked wing aircraft.

The inverse problem of determining the supersonic flow past a thin symmetrical wing at zero incidence with given surface pressure distribution is solved for wings of arbitrary plan form. Expressions are obtained for the vertical derivative of the potential on the wing surface from which, using the linearised boundary condition of zero normal velocity at the surface, the profile of the wing can be designed. The integral equation method adopted by J. C. Evvard and extended by G. N. Ward is used. The analysis cannot be applied to pointed wings, whose leading edges are subsonic. The results in Part I are completely general and are applied to specific problems in Part II.

The flow past an airscrew rotating with uniform angular velocity in a uniform supersonic stream is considered from the point of view of axes fixed in the airscrew, the motion relative to these axes being steady. A linearised equation and boundary conditions are found for the potential of the disturbance flow caused by airscrews which produce only small perturbations in the main stream. This equation is solved by expanding the potential in powers of the ratio, χ, of the tip speed of the airscrew to the speed of sound in the undisturbed stream. Equations and boundary conditions are found for the coefficients of each term of the series and it is seen that the term independent of χ satisfies the ordinary linearised potential equation for fixed axes. By subtracting suitable special integrals, the more complicated equations for the coefficients of the various powers of χ can be reduced to this ordinary potential equation. Hadamard's methods have been applied by Puckett, Evvard and, finally, Ward to the problem of finding the flow past a thin wing fixed in a supersonic stream and their methods are applied to find the terms of the series in our case. The first two terms are found explicitly for a particular type of windmill. Approximate expressions are then developed for the torque and the drag on such a windmill and the results applied in special cases. The best working conditions for windmills of this type are found by considering the effect on efficiency and power of varying the shape and the speed of rotation, within the limits imposed by the strength of the material of the airscrew.

The design of ducted fans, and the estimation of their performance for conditions other than those of the design, are usually based on the equations of the so-called strip-theory. Existing methods of solution invariably employ some form of iteration which is more or less rapidly convergent according to the experience of the computer and the detail of the method.

In this note, a solution is given which, although it is strictly an iteration, is explicit in the sense that in practice no iteration is necessary. It is shown how both design for a given performance and performance for a given fan can be calculated in one direct step.

Conditions for maximum efficiency are considered in an Appendix. Here a suggestion of R. Baldur's is used to give a simple approximate solution of the strip-theory equations which can be developed to give conditions for maximum efficiency, for the special case of a constant pressure rise through the fan.

The design of the damped tuned vibration absorber, which has previously been treated algebraically, is studied by the use of a three-dimensional admittance diagram. The advantages of this method are that the basic principles can be more easily understood and that it is not necessary to simplify the vibrating system so much. The results obtained by the use of the method show that under some conditions the algebraic treatment does not give the optimum tuning, but that the differences in performance of dampers which have been designed by the two methods will not be large. It is shown how the method may be extended to more complex problems.

The elementary engineering theory of longitudinal waves in beams allows for no dispersion, whereas Love's corrected version of it predicts that very short waves are not propagated. A new correction is added which overcomes these defects. To do this, a simple theorem is proved concerning certain types of closed region. A method of improving Love's formula and the present one is illustrated.

Wild has extended to three dimensions the von Kármán-Pohlhausen method for the laminar boundary layer flow over a fixed obstacle and used the method for an infinite yawed elliptic cylinder in a stream. In this paper the method is tested in two ways (which may be called full-Pohlhausen and semi-Pohlhausen) for the case of an infinite yawed cylinder when the velocity outside the boundary layer over the surface normal to the generators is of the form U=cxm. The exact solution is known in this case.

A table of the skin friction, displacement thickness and momentum thickness is given for various values of β [=2m/(m + 1)], and the agreement is found to be fairly good for β>0 (accelerated flow) but not so good for β<0 (retarded flow).)

The method developed by G. N. Ward for the treatment of slender pointed bodies in a uniform supersonic stream is applied to three special cases.

• (i) Supersonic flow past a body of revolution with thin wings of symmetrical section and of small aspect ratio at zero incidence.

• (ii) Supersonic flow past a body of revolution with plane wings of small aspect ratio set at incidence to the body, the whole being at incidence to the stream.

• (iii) Supersonic flow past a body of revolution with a plane fin of small aspect ratio set at incidence, the whole being at incidence to the stream.

The pressure distribution on the wing has been calculated for a special case of (i) and is given in the Appendix.

The general technique for rendering approximate solutions to physical problems uniformly valid is here applied to the simplest form of the problem of correcting the theory of thin wings near a rounded leading edge. The flow investigated is two-dimensional, irrotational and incompressible, and therefore the results do not materially add to our already extensive knowledge of this subject, but the method, which is here satisfactorily checked against this knowledge, shows promise of extension to three-dimensional, and compressible, flow problems.

The conclusion, in the problem studied here, is that the velocity field obtained by a straightforward expansion in powers of the disturbances, up to and including either the first or the second power, with coefficients functions of co-ordinates such that the leading edge is at the origin and the aerofoil chord is one of the axes, may be rendered a valid first approximation near the leading edge, as well as a valid first or second approximation away from it, if the whole field is shifted downstream parallel to the chord for a distance of half the leading edge radius of curvature ρL. It follows that the fluid speed on the aerofoil surface, as given on such a straightforward second approximation as a function of distance x along the chord, similarly is rendered uniformly valid (see equation (52)) if the part singular like x-1 is subtracted and the remainder is multiplied by .

The integral equations for the pressure distribution on an oscillating thin wing having a prescribed velocity distribution in a supersonic flow are developed.

The theory of aerofoils in unsteady flow, which has made substantial progress in the last decade due largely to the ground work of Küssner and his co-workers, is presented here in a form suitable for application in aeroelastic problems, particularly those concerned with the dynamic loads on aircraft arising from gusts.

Exact expressions are given for the aerodynamic lift and moment for an oscillating aerofoil as well as for the case of arbitrary motion through disturbed air. The expressions for the latter case involve two special functions, generally referred to as Wagner and Küssner functions. Exact values of these functions are tabulated together with useful approximations. The problem of a wing-tail combination is discussed and a method of solution indicated. The bibliography at the end of the paper lists the most important publications in this field.

A method is developed for calculating the profile drag of a yawed wing of infinite span, based on the assumption that the form of the spanwise distribution of velocity in the boundary layer, whether laminar or turbulent, is insensitive to the chordwise pressure distribution. The form is assumed to be the same as that accepted for the boundary layer on an unyawed plate with zero external pressure gradient. Experimental evidence indicates that these assumptions are reasonable in this context. The method is applied to a flat plate and the N.A.C.A. 64-012 section at zero incidence for a range of Reynolds numbers between 106 and 108, angles of yaw up to 45°, and a range of transition point positions. It is shown that the drag coefficients of a flat plate varies with yaw as cos½ Λ (where Λ is the angle of yaw) if the boundary layer is completely laminar, and it varies as if the boundary layer is completely turbulent. The drag coefficient of the N.A.C.A. 64-012 section, however, varies closely as cos½ Λ for transition point positions between 0 and 0.5 c. Further calculations on wing sections of other shapes and thicknesses and more detailed experimental checks of the basic assumptions at higher Reynolds numbers are desirable.

In a previous paper it was assumed that the cross-girders are freely supported at their ends. In practice these ends can often be considered as clamped (“ built-in ”), for example, a span of continuous bottom-girders of a ship between two transverse bulkheads.

In the following discussion it is assumed that the ends of the cross-girders are clamped, conditions at the ends of the main beams remaining arbitrary, as in Ref. 1.