CoA. Reports

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Open Access
On the analysis of statically indeterminate structures
(College of Aeronautics, Cranfield, 1946-11) Hemp, W. S.
This report develops a general method for the analysis of statically indeterminate structures. It concerns itself both with a rigorous demonstration of the validity of the methods and with recommendations for their successful practical application. Continues ...
Open Access
Ignoration of distortional co-ordinates in the theory of stability and control
(College of Aeronautics, 1946-12) Duncan, W. J.
Gates and Lyon have proposed to treat theoretically the stability and control of deformable aircraft by a method in which the distortion co-ordinates are ignored and the influence of distortion is allowed for by suitable modifications of the derivatives and other coefficients. in the 'present paper an exact method for eliminating the distortion co-ordinates is given and the conditions in which the true eliminant conforms with the simplification of Gates and Lyon are examined. In general the simplification is not justified mathematically, but in certain circumstances it provides an acceptable approximation. it will not be practically valid unless the structural distortions occur so relatively slowly that the associated inertia forces are negligible, i.e. the distortions must be quasi-static.
Open Access
Note on the application of the linearised theory for compressible flow to transonic speeds
(College of Aeronautics, Cranfield, 1947-01) Robinson, A.; Young, A. D.
It is shown that for finite aspect ratio the linearised theory of compressible flow remains theoretically consistent in the region of transonic speeds, although tis predictions may deviate appreciably from experimental results in that region. The variation of the theoretical lift curve slope of an aerofoil of finite span is considered as the mach number increases from below unity to above unity, and it is shown that the lift curve slope remains finite and continuous.
Open Access
Technique of the step-by-step integration of ordinary differential equations
(College of Aeronautics, Cranfield, 1947-02) Duncan, W. J.
In Part 1 step-by-step methods are examined critically and emphasis is placed on the dependence of the error on the number n of steps used for a given range of the independent variable. The index of a process is defined and it is shown that the errors can be assessed and partially corrected when the index is known and results obtained for two or more values of n. Attention is drawn to the advantages in certain cases of a part-analytical process. In Part 2 methods of numerical integration in general are classified and briefly reviewed. The chart, Table 2.2.1, summarises the classification.
Open Access
Note on the dynamics of a slightly deformable body
(College of Aeronautics, Cranfield, 1947-02) Hemp, W. S.
The purpose of this note is to develop the equations of motion of a slightly deformable body. Appeal to general principles shows the independence of the translation (Para.2). Moving axis are defined in papa.3 which can be taken to define the rotation. Motion relative to these axes is described by normal co-ordinates (Para.4) and the kinetic energy of the motion relative to the centre of mass is split into two parts; the energy of rotation and the energy of vibration (Para.5). Equations for the vibration are then formulated (Para.6). Attention is drawn to the coupling between rotation and vibration, which only vanishes when the angular velocities are small (Para.7).
Open Access
The effect of the sweepback of delta wings on the performance of an aircraft at supersonic speeds
(College of Aeronautics, Cranfield., 1947-03) Robinson, A.; Davies, F.T.
The variation with sweepback of total drag of an aircraft in level flight at supersonic speeds is calculated. It is shown that sweepback is not uniformly beneficial, but that in general the optimum amount of sweepback depends in the design speed and altitude.
Open Access
Wing body interference at supersonic speeds
(College of Aeronautics, Cranfield, 1947-04) Kirkby, S.; Robinson, A.
The increment in lift due to wing-body interference at supersonic speed is calculated approximately for an untapered wing without sweepback.
Open Access
Note on the velocity and temperature distributions attained with suction on a flat plate of infinite extent in compressible flow
(College Of Aeronautics, Cranfield, 1947-08) Young, A. D.
The problem considered by Griffith and Meredithl for incompressible flow is here considered for compressible flow, it being assumed that there is no heat transfer by conduction at the plate. Essentially, the method consists of establishing a correspondence between the velocity and temperature profiles for incompressible flow and those for compressible flow, the lateral ordinated being scaled by factors which are functions of the ordinates and of Mach number. The results of calculations covering a range of Mach numbers up to 5.0 are shown in Figs. A and 2.
Open Access
Application of the linear perturbation theory to compressible flow about bodies of revolution
(College of Aeronautics, Cranfield, 1947-09) Young, A. D.; Kirkby, S.
The linearised theory is developed in some detail in order to clarify the differences between two-dimensional and axi-symmetric flow. In agreement with other authors it is concluded that the perturbation velocity on a thin body of revolution in compressible flow is 1/β2 times the perturbation velocity in incompressible flow on a thinner body at reduced incidence obtained by reducing the lateral dimensions of the original body in the ratio (3:1). This result is applied to a representative family of streamline bodies of revolution at zero incidence. Continues…
Open Access
Bound and trailing vortices in the linearised theory of supersonic flow, and the downwash in the wake of a delta wing
(College of Aeronautics, Cranfield, 1947-10) Robinson, A.; Hunter-Tod, J. H.
The field of flow round a flat aerofoil at incidence can/may be regarded in linearised theory as the result of both bound and trailing vortices for supersonic as well as for low speed flight. This leads to a convenient method, given the lift distribution over an aerofoil, for calculating the flow round it at supersonic speeds. As an application of the results the downwash is calculated in the wake of a delta wing lying within the Mach cone emanating from its apex. The downwash is found to be least just aft the trailing edge and is everywhere less that the downflow at the aerofoil. It increases steadily to a limiting value which is attained virtually within two chord lengths of the trailing edge. The ratio of the downwash at any point in the wake to the downflow at the aerofoil decreases with increasing Mach number and apex angle.
Open Access
Source and vortex distributions in the linearised theory of steady supersonic flow
(College of Aeronautics, Cranfield, 1947-10) Robinson, A.
The hyperbolic character of the differential equation satisfied by the velocity potential in linearised supersonic flow entails the presence of fractional infinities in the fundamental solutions of the equation. Difficulties arising from this fact can be overcome by the introduction of Hadmard’s ‘finite part of an infinite integral’. Together with the definition of certain counterparts of the familiar vector operators this leads to a natural development of the analogy between incompressible flow and linearised supersonic flow. In particular, formulae are derived for the field of flow due to an arbitrary distribution of supersonic sources and vortices. Applications to Aerofoil theory, including the calculation of the downwash in the wake of an aerofoil, are given in a separate report.
Open Access
Assessment of errors in approximate solutions of differential equations
(College of Aeronautics, Cranfield, 1947-12) Duncan, W. J.
The term assessment is applied to any process which enables us to set rigid bounds to the error or to estimate its value. It is shown that upper and lower bounds can be assigned whenever the Green's function of the problem is one- signed; this is true in many important problems. Another method is applicable to step by step -solutions of ordinary differential equations, linear or non-linear, and depends on use of the "index" of the process of integration. Lastly, the error in a linear problem can be estimated when an approximation to the Green's function is known.
Open Access
The aerodynamic derivatives with respect to sideslip for a delta wing with small dihedral at supersonic speeds
(College of Aeronautics, 1947-12) Robinson, A.; Hunter-Tod, J. H.
Expressions are derived for the sideslip derivatives on the assumptions of the linearised theory of flow for a delta wing with small dihedral flying at supersonic speeds. A discussion is included in the appendix on the relation between two methods that have been evolved for the treatment of aerodynamic force problems of the delta wing lying within its apex Mach cone. Continues…
Open Access
Skin friction in the laminar boundary layer in compressible flow
(College of Aeronautics, 1948) Young, A. D.
Open Access
On a theory of sandwich construction
(College of Aeronautics, 1948-03) Hemp, W. S.
The theory of sandwich construction developed in this paper proceeds from the simple assumption that the filling has only transverse direct and shear stiffnesses, corresponding to its functional requirements (§1). This supposition permits integration of the equilibrium equations for the filling (§2). The resulting integrals are used to study the compression buckling of a flat sandwich plate (§3). The formulae obtained are complex, but may be simplified in practical cases (§4). A second approach to sandwich problems is made in §5, where a theory of "bending" of plates is outlined. This generalises the usual theory, making allowance for flexibility in sheer. This approach is applied to overall compression buckling of a plate in §6, and agreement with the previous calculations is found. This suggests the possibility of calculating buckling loads for curved sandwich shells. A simple example, the symmetrical buckling of a circular cylinder in compression is worked out in §7. The theory developed would seem applicable to all cases of buckling of not too short a wave length (§8).
Open Access
Note on the limits to the local Mach number on an aerofoil in subsonic flow
(College of Aeronautics, 1948-04) Young, A. D.
It has been noted in some experiments that the local Mach number just ahead of a shock wave on an aerofoil in subsonic flow is limited, values of the limit of the order of 1.4 are usually quoted. This note presents two lines of thought indicating how such a limit may arise. The first starts with the observation that the pressure after the shock will not be higher than the rain stream pressure. Fig.1 shows the calculated relation between local Mach number ahead of the shock (M„ 1 ), shock inclination (S), mainstream Mach number (M1) and pressure coefficient just aft of the shock. • (Cp) It is noted that, for given M1 , Cp and .5 ,two shocks are possible in general, a strong one for which Ms , > 1.48, and a weak one for which MS1 < 1.48, and it is argued that the latter is the more likely. The second approach is based on the fact that a relation between stream deflection (8) and Mach number for the flow in the limited supersonics regions on a number of aerofoils has been derived from some. experimental data. Further analysis of experimental data is required before this relation can be accepted as general. If it is accepted, however, then it indicates that the Mach numbers increase above unity for a given deflection is about one-third of that given by simple wave theory (Fig.2). An analysis of the possible deflections on aerofoils of various thicknesses (Fig.3) then indicates that deflections corresponding to local Mach numbers of the order of 1,5 or higher are unlikely except at incidences of the order of5 ° or more, and may then be more likely for thick wings than for thin wings. Flow breakaway will make the attainment of such high local Mach numbers less likely.
Open Access
The relative accuracy of quadrature formulae of the Cotes' closed type
(College of Aeronautics, Cranfield, 1948-05) Kirkby, S.
Quadrature formulae, such as those discovered by Gregory, Newton, Simpson and Cotes, which are derivable by integration of Lagrange’s interpolation formula between definite limits, are classified as Cotes’ Type Formulae. When the functional values at the end –points of the range of integration are used the corresponding formulae are said to be of the ‘closed type’. It is shown that, for closed type formulae, the error due to application of a 2n-strip formula is in general less than that due to a (2n+a) –strip formula over the same range of integration when using the same tabular interval of the argument.
Open Access
On some problems of unsteady supersonic aerofoil theory
(College of Aeronautics, Cranfield, 1948-05) Robinson, A.
Unsteady supersonic flow round an aerofoil of infinite span is considered in the first part of the paper.It is shown that the pressure at any given point of an aerofoil under forward acceleration can be analysed into three components, one of which is the steady (Ackeret)pressure due to the instantaneous velocity, while of the other two, one depends directly on the acceleration, and one on the square of the velocity, during a limited time interval preceding the instant under consideration. However, the difference between the total pressure and the "steady pressure component" is such that it can be neglected in all the definitely supersonic conditions which are likely to occur in practice. The oscillatory supersonic flow round a Delta wing inside the Mach cone emanating from its apex is considered in the second part of the paper. Particular "normal" solutions are obtained by means of a special system of curvilinear coordinates. It is shown that the velocity potentials corresponding to vertical and pitching oscillations of the wing can be represented by series of such normal solutions. The assumptions of linearised theory arc adopted throughout.
Open Access
On the integration of hyperbolic differential equations
(College of Aeronautics, Cranfield, 1948-07) Robinson, A.
See full text for abstract.
Open Access
Flutter of systems with many freedoms
(College of Aeronautics, Cranfield, 1948-08) Duncan, W. J.
Experience has shown that it is often necessary to retain many degrees of freedom in order to calculate critical flutter speeds reliably, but this entails much labour. Part 1 discusses the choice of a minimum set of freedoms and suggests that this should be based on the equation of energy and the use of the Lagrangian dynamical equation corresponding to any proposed additional freedom. The method for conducting flutter calculations so as to minimise labour are treated in Part 2.