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THE continual development of helicopter rotor systems has so far resulted in the use of about six main types, and it will be of value briefly to recapitulate their advantages and disadvantages in order to obtain a balanced picture against which the stiff‐hinged rotor can be judged.

THE simple actuator disk theory, first postulated by Froude over sixty years ago, is the basis of most helicopter induced flow theory. This disk is an idealization of a rotor which uniformly accelerates the air with no loss of thrust at the blade tips. It can therefore be regarded as the limit case of a rotor with an infinite number of blades. It is also assumed to be infinitely thin so that no discontinuities in velocity occur on the two sides of the disk.

In‐plane vibration of a balanced helicopter rotor is caused by variations with azimuth of the in‐plane forces acting on individual blades. These forces may be summarized under three headings: ‘Induced forces’ caused by the inclination of elemental lift vectors relative to the axis of rotation. ‘Profile drag forces’: variations are caused by changes with azimuth angle of the angle and airspeed of the individual blade elements. ‘Coriolis forces’, which are caused by blade flapping, which brings about a variation of blade moment of inertia about the axis of rotation. Equations are developed in this paper for the resultant hub force due to each of these forces, on the assumptions of small flapping hinge offset. It is assumed that blades are linearly twisted and tapered, an assumption which in practice can be applied to any normal rotor. It is shown that by suitably inclining the mechanical axis it is possible to balance out the worst induced and profile drag vibrations by the coriolis one, which can be made to have opposite sign. If the mechanical axis is fixed in the fuselage, this suppression is fully effective for one flight condition only. In multi‐rotor helicopters, vibration suppression can be extended over a much wider range by varying the fuselage attitude. The logical result of this analysis is, for single rotor helicopters, a floating mechanical axis which can be adjusted or trimmed by the pilot. This would be quite simple to do on a tip‐driven rotor, and has already been achieved with a mechanical drive on the Doman helicopter. The more important causes of vibration from an unbalanced rotor are next con‐sidered, attention here being confined principally to fully articulated rotors, which are the most difficult to balance because the drag hinges tend to magnify all in‐accuracies in finish and balance. From a brief discussion of the vertical vibration of an imperfect rotor it is shown that some contemporary methods of ‘tracking’ are fundamentally wrong. Finally the vibration due to tip‐mounted power units is described. In discussing the effect of a vibratory force on a helicopter a simple response chart is developed, and it is thought that its use could well be accepted as a simple standard for general assessment purposes. In the development of equations for vibration the following points of general technical interest are put forward: An equation for induced torque is developed which includes a number of hitherto neglected parameters. A new form of equation for mean lift coefficient of a blade is suggested. The simple Hafner criterion for flight envelopes is shown to give rise to considerable error, and the use of Eq. (28) is suggested in its place. The variation of profile torque with forward speed is given, and the increase due to ? varying round the disk is expressed as an explicit equation, thus allowing considerable improvement in the present methods of allowing for this effect.

A theory is developed to describe the dynamic behaviour, control angles to trim and stability derivatives of an aerodynamic servo‐controlled rotor. The analysis is restricted to constant chord rotor blades which are torsionally deformed by the servo flap to give changes in rotor pitch angle, as this is the form in which the system is most likely to be used. Also, the aerodynamic centre and elemental C.G. lines are assumed to coincide with the blade torsion axis. Since the stiff hinge assumption is used, the analysis is applicable to blades with offset or ‘stiff’ flapping hinges, or to cantilever rotors, which can be simulated by a rigid blade with a stiff hinge. Comparison with some N.A.C.A. test tower results shows that the theory developed gives excellent agreement with the available experimental results. In Appendix I the stability of the tip path plane is examined, using the equations derived in the report, and three regions of instability are shown to be present. A practical rotor must be designed to operate below the lowest instability region, as is the case for the N.A.C.A. test rotor. Equations for ∂a1s/∂µ and ∂a1s/∂q are developed for the low speeds near hovering in Appendix II. Other derivatives can be easily derived from the general equations of motion given in Table 1.

In Part 1 it is shown that the equation for blade equilibrium about the flapping pin is

A NUMBER of approaches to the calculation of rotor downwash have already been discussed. Broadly spsaking, the methods of Castles and DeLeeuw and Squire and Mangler are the same. In both methods the downwash at the rotor disk is assumed to be perpetrated in a helical downwash sheet which, as the slipstream, extends below the rotor to infinity. The downwash in the disk due to the bound vortices, and the additional downwash in the disk which is induced by the helical sheets in the slipstream (Castles and DeLeeuw substitute downwash rings for helices, in the interest of mathematical simplicity) is calculated, on the assumption of an infinite number of lightly loaded blades. The final results of Castles and DeLeeuw on the one hand, and Squire and Mangier on the other, are in very wide disagresment. This disagreement is principally due to the fact that, whereas the first investigation assumes constant circulation along the blade (ideal twist and taper), Mangier and Squire assume a ‘practical’ variation of the form likely to be encountered on an untwisted untapered blade. We conclude that the radial distribution of lift on a helicopter blade will have a profound effect on the downwash pattern: which in turn will affect the calculated lift.

The theory of rotor dynamics given in Ref. 1 is extended to include the effects of coupling between feathering and flapping (δ3 angle) and flapping hinge offset. Both introduce considerable modification to the classic equations, and instead of simple explicit equations for flapping amplitudes, coning angle, collective pitch and inflow angles, five simultaneous equations have now to be solved. Data sheets have been constructed which enable this to be done quickly and accurately for any design of linearly tapered and twisted blade. It is suggested that the intelligent use of such data sheets is of great assistance in a design office, not only because of the very considerable time savings achieved, but also because they eliminate the most fruitful sources of error in numerical calculation. It is shown that a high offset rotor enables much higher speeds to be achieved with a conventional helicopter—an effect which has already been fairly well publicized. A penalty is paid for this in the form of hub pitching moments which have to be balanced out externally; either by the use of two rotors, offset C.G., aerodynamic surfaces, or inclination of the mechanical axis. These effects will be considered in detail in a further article. Finally, equations are developed for a convenient method of calculating blade elemental angle of attack which is claimed to be superior to classic methods for design office purposes.

The elimination of the retreating blade stall speed limitation for helicopters by means of an appropriately programmed feathering input is studied for the general case of a rigid flapping blade with hinge constraint (thus making the results applicable to conventional, offset‐hinged or cantilevered rotor blades). It is concluded that second harmonic feathering alone will not be particularly effective in delaying the stall limit, but that a suitable programme of several higher harmonic inputs will enable the retreating blade stall limit to be pushed beyond the advancing blade compressibility limit. In the course of the investigation generalized equations were developed for blade flapping to the nth harmonic under the influence of feathering to the nth harmonic. The resultant matrix is symmetrical and checks with the few available limit cases derived by other workers. Because of loose coupling in the matrix generalized equations can be derived giving the effect of any particular harmonic of feathering upon flapping and angle of attack distribution around the disk. The effect of higher harmonic feathering upon rotor stability derivatives is not discussed in this text, but examination of the equations indicates that an improvement in stability could be obtained by the application of second harmonic control. This paper does not discuss the mechanical details of obtaining a higher harmonic feathering input, nor is it suggested that this is necessarily the best means of obtaining higher forward speeds. In certain cases it may be the only means however.

In Ref. (1) the dynamics of a rotor with high offset flapping hinges were considered, for twisted and tapered blades with ?3 coupling between flapping and feathering. In the present paper simple expressions are derived for the hub pitching and rolling moments of such a rotor, and it is shown that these moments must be balanced either externally (by aerodynamic surfaces for example) or internally by suitable inclination of the mechanical axis (Eqs. (7) and (8)). The relevant expression for total rotor moment is given in Eq. (17). In order to arrive at a solution, the vertical force acting on a single flapping pin was evaluated, and this is given by Eq. (5). This is particularly interesting, apart from stress calculations, in that it shows the mechanism whereby a blade which is out of track in the tip path plane can cause a first harmonic vertical vibration in forward flight. It is shown that a two‐bladed high offset rotor will always experience second harmonic hub vibration unless it is balanced out by inclination of the mechanical axis. In general, the mechanical axis position is seen to be an important parameter in both vibration and hub moments; a fact which has hitherto passed unrecognized except for some unpublished work at the Royal Aircraft Establishment by Mr P. Brotherhood. This does not necessarily imply that mechanical axis inclination is a panacea to cure all ills. By far its most useful effect lies in suppressing in‐plane vibration, to which vertical vibration is usually of second order importance. This is, of course, a separate subject.

AT low forward speeds the slipstream from a helicopter rotor is substantially downwards in direction and will cause a drag force to be generated on any body immersed in it, the drag acting in the direction of the slipstream. In most performance methods the effect of this vertical drag is ignored, bat it cart in fact substantially modify calculated performance, being equivalent to a weight increase of over 10 per cent even on some single rotor designs. The basic parameter is the equivalent flat plate area (area of body drag coefficient) which is immersed in the slipstream, and this is expressed as a ratio of the rotor disk area, i.e. ACD/πR2.