Flapping motion

Flapping motion

To examine the flapping motion more fully we assume, unless otherwise stated, that the flapping hinge is on the axis of rotation. This simplifies the considerations without hiding anything of significance. Referring to Fig 4.12, the flapping takes place under conditions of dynamic equilibrium, about the hinge, between the aerodynamic lift (the exciting function), the centrifugal force (the ‘spring’ or restraining force) and the blade inertia (the damping). In other words, the once-per-cycle oscillatory motion is that of a dynamic system in resonance. The flapping moment equation is seen to be


We shall return to this equation later. The centrifugal force is by far the largest force acting on the blade and provides an essential stability to the flapping motion. The degree of stability is highest in the hover condition (where the flapping angle  is constant) and decreases as the advance ratio increases. Bramwell’s consideration of the flapping equation (p. 153 et seq.) leads in effect to the conclusion that the motion is dynamically stable for all realistic values of m. Maximum flapping velocities occur where the resultant air velocity is at its highest and lowest, that is at 90° and 270° azimuth. Maximum displacements occur 90° later, that is at 180° (upward) and 0° (downward). These displacements mean that the plane of rotation of the blade tips,the tip-path plane(TPP),is tilted backwards relative to the plane normal to the rotor shaft, the shaft normal plane (SNP). In hover the blades cone upwards at a constant angle a0, say, to the shaft normal plane. The coning angle is that at which the blade weight is supported by the aerodynamic lift. Its existence has an additional effect on the orientation of the TPP during rotation in forward flight. Figure 4.13 shows that because of the coning angle, the flight velocity V has a lift-increasing effect on a blade at 180° (the forward blade) and a lift-decreasing effect on a blade at 0° (the rearward blade). This asymmetry in lift is, we see, at 90° to the side-to-side asymmetry discussed earlier: its effect is to tilt the TPP laterally and since the point of lowest tilt follows 90° behind the point of lowest lift, the TPP is tilted downwards to the right, that is on the advancing side. The coning and disc tilt angles are normally no more than a few degrees. Since in any steady state of the rotor the flapping motion is periodic, the flapping angle can be expressed in the form of a Fourier series:

Rotor Mechanisms for Forward Flight

Rotor Mechanisms for Forward Flight

 The edgewise rotor


In level forward flight the rotor is edgewise on to the airstream,a basically unnatural state for propeller functioning. This is shown in Fig. 4.1. Practical complications which arise from this have been resolved by the introduction of mechanical devices, the functioning of which in turn adds to the complexity of the aerodynamics. Figure 4.2 pictures the rotor disc as seen from above. Blade rotation is in a counter-clockwise sense (the standard adopted for all helicopters of the Western countries) with rotational speed W. Forward flight velocity is V and the ratio V/WR, R being the blade radius, is known as the advance ratiosymbol m,and has a value normally within the range zero to 0.5. Azimuth angle y is measured from the downstream blade position: the range y=0°–180° defines the advancing side and that from 180°–360° (or 0°) the retreating side. A blade is shown in Fig 4.2 at 90° and again at 270°. These are the positions of maximum and minimum relative air velocity normal to the blade, the velocities at the tip being (WR + V) and (WR - V), respectively. If the blade were to rotate at fixed incidence, then owing to this velocity differential, much more lift would be generated on the advancing side than on the retreating side. Calculated pressure contours for a fixed-incidence rotation with m=0.3 are shown in Fig 4.3. About four-fifths of the total lift is produced on the advancing side. The consequences of this imbalance would be large oscillatory bending stresses at the blade roots and a large rolling moment on the vehicle. Both structurally and dynamically the helicopter would be unflyable.

Example of hover characteristics

Example of hover characteristics

Corresponding to CL/a and CD/CL characteristics for fixed wings, we have CT/qand Cp/CT for the helicopter in hover. An example has been evaluated using the following data: blade radius, R = 6m blade chord (constant), c = 0.5m blade twist, linear from 12° at root to 6° at tip number of blades, N = 4 empirical constant, k=1.13 blade profile drag coefficient (constant), CD0 = 0.010 The variation of CT/s with q is shown in Fig. 3.8(a). The nonlinearity results from the CT term in Equation (3.28). The variation of CP/s with q is calculated for three cases:

• k=1.13, Equation (3.48), • k=1.0, Equation (3.46), the simple momentum theory result, • Figure of merit M = 1.0, which assumes k=1.0 and CDo = 0.

Over the range shown (Fig. 3.8(b)), using the factor k=1.13 results in a power coefficient 0–9% higher than that obtained using simple momentum theory. The curve for M = 1 is of course unrealistic but gives an indication of the division of power between induced and profile components. (Rotor performance characteristics are sometimes plotted as CP/s versus CT/s. This type of plot is known as a hover polar.)

Tip loss

Tip loss

A characteristic of the actuator disc concept is that the linear theory of lift is maintained right out to the edge of the disc.Physically,recalling Fig.2.1a–2.1c,we suppose the induced velocity,in which the pressure is above that of the surrounding air, to be contained entirely below the disc in a well-defined streamtube surrounded by air at rest relative to it. In reality, because the rotor consists of a finite number of separate blades, some air is able to escape outwards between the tips, drawn out by the tip vortices. Thus the total induced flow is less than the actuator disc theory would prescribe,so that for a given pitch setting of the blades the thrust is somewhat lower than that given by Equation (3.22). The deficiency is known as tip loss and is shown by a rapid falling off of lift over the last few per cent of span near the tip, in a blade loading distribution such as that of Fig. 2.13. Although several workers have suggested approximations [Bramwell (p. 111) quotes Prandtl, Johnson (p. 60) quotes in addition Sissingh and Wheatley] no exact theory of tip loss is available. A common method of arriving at a formula is to assume that outboard of a station r = BR the blade sections produce drag but no lift. Then the thrust integral in Equation (3.21) is replaced by

Blade mean lift coefficient

Blade mean lift coefficient

Characteristics of a rotor obviously depend on the lift coefficient at which the blades are operating and it is useful to have a simple approximate indication of this. The blade mean lift coefficient provides such an indication. As the name implies the mean lift is that which, applied uniformly along the blade span, would give the same total thrust as the actual blade. Writing the mean lift coefficient as L we have, from Equation (3.12),

from which


The parameter CT/s is thus of fundamental importance and this explains the preference some workers have for using it as the definition of thrust coefficient (see Equation (3.18)) instead of CT. Expanding the definition gives

Ideal twist

Ideal twist

The relation in Equation (3.33) contains one particular case when l is indeed constant, namely if qr is constant along the span, that is qt being the pitch angle at the tip. This non-linear twist is not physically realizable near the root but the case is of interest because, as momentum theory shows, uniform induced velocity corresponds to minimum induced power. The analogy with elliptic loading for a fixed-wing aircraft is again recalled. The twist in Equation (3.34) is known as ideal twist. Inserting in Equation (3.21) gives

Non-uniform inflow A questionable assumption which has been made so far is that the induced velocity is uniform across the blade span. The effect of nonuniformity can be allowed for by using differential forms of the appropriate equations in the combination of blade element theory and momentum theory. Equation (3.21) in the blade element theory is replaced by

 Non-uniform inflow

A questionable assumption which has been made so far is that the induced velocity is uniform across the blade span. The effect of nonuniformity can be allowed for by using differential forms of the appropriate equations in the combination of blade element theory and momentum theory. Equation (3.21) in the blade element theory is replaced by  which expresses the element of thrust on an annulus of the disc at radius r. The corresponding equation from momentum theory, again using the hover case for simplicity, is the replacement of Equation (2.2), namely.

Rotor in Vertical Flight: Blade Element Theory

Rotor in Vertical Flight: Blade Element Theory

Basic method

Blade element theory is basically the application of the standard process of aerofoil theory to the rotating blade.A typical aerodynamic strip is shown in Fig. 3.1 and the appropriate notation for a typical strip is shown in Fig. 3.2. Although in reality flexible, the blade is assumed throughout to be rigid, justification for this lying in the fact that at normal rotation speeds the outward centrifugal force is the largest force acting on a blade and in effect is sufficient to hold the blade in rigid form. In vertical flight, including hover, the main complication is the need to integrate the elementary forces along the blade span. Offsetting this, useful simplification occurs because the blade incidence and induced flow angles are normally small enough to allow small-angle approximations to be made. Fig. 3.3 is a plan view of the rotor disc, viewed from above. Blade rotation is anticlockwise (the normal system in Western-world countries) with angular velocity W.The blade radius is R,the tip speed therefore being WR, alternatively written as Vt. An elementary blade section is taken at radius y, of chord length c and spanwise width dy. Forces on the blade section are shown in Fig. 3.4. The flow seen by the section has velocity components Wy in the disc plane and (vi + Vc) perpendicular to it. The resultant of these is The blade pitch angle, determined by the pilot’s collective control setting (see Chapter 4), is q. The angle between the flow direction and the plane of rotation, known as the inflow angle is f, given by 

Wake analysis methods

 Wake analysis methods

By analysis of his carefully conducted series of smoke-injection tests, Landgrebe5 reduced the results to formulae giving the radial and axial coordinates of a tip vortex in terms of azimuth angle, with corresponding formulae for the inner sheet. From these established vortex positions the induced velocities at the rotor plane may be calculated. The method belongs in a general category of prescribed-wake analysis, as do earlier analyses by Prandtl, Goldstein and Theodorsen, descriptions of which are given by Bramwell. These earlier forms treated either a uniform vortex sheet as pictured in Fig. 2.11 or the tip vortex in isolation, and so for practical application are effectively superseded by Landgrebe’s method. More recently, considerable emphasis has been placed on free-wake analysis, in which modern numerical methods are used to perform iterative calculations between the induced velocity distribution and the wake geometry, both being allowed to vary until mutual consistency is achieved. This form of analysis has been described for example by Clark and Leiper6. Generally the computing requirements are very heavy, so considerable research effort also goes into devising simplified free-wake models which will reduce the computing load. Calculations for a rotor involve adding together calculations for the separate blades. Generally this is satisfactory up to a depth of wake corresponding to at least two rotor revolutions. A factor which helps this situation is the effect on the tip vortex of the upwash ahead of the succeeding blade – analogous to the upwash ahead of a fixed wing. The closer the spacing between blades, the stronger is this effect from a succeeding blade on the tip vortex of the blade ahead of it; thus it is observed that when the number of blades is large, the tip vortex remains approximately in the plane of the rotor until the succeeding blade arrives, when it is convected downwards. In the ‘far’ wake, that is beyond a depth corresponding to two rotor revolutions, it is sufficient to represent the vorticity in simplified fashion; for example free-wake calculations can be simplified by using a succession of vortex rings, the spacing of which is determined by the number of blades and the mean local induced velocity. Eventually in practice both the tip vortices and the inner sheets from different blades interact and the ultimate wake moves downward in a confused manner. 

Complexity of real wake

Complexity of real wake

The actuator disc concept, taken together with blade-element theory, serves well for the purposes of helicopter performance calculation. When, however, blade loading distributions or vibration characteristics are required for stressing purposes it is necessary to take into account the real nature of flow in the rotor wake. This means abandoning the disc concept and recognizing that the rotor consists of a number of discrete lifting blades, carrying vorticity corresponding to the local lift at all points along the span.Corresponding to this bound vorticity a vortex system must exist in the wake (Helmholtz’s theorem) in which the strength of wake vortices is governed by the rate of change of circulation along the blade span. If for the sake of argument this rate could be made constant, the wake for a single rotor blade in hover would consist of a vortex sheet of constant spanwise strength, descending in a helical pattern at constant velocity, as illustrated in Fig. 2.11. The situation is analogous to that of elliptic loading with a fixed wing, for which the induced drag (and hence the induced power) is a minimum. This ideal distribution of lift, however,is not realizable for the rotor blade, because of the steadily increasing velocity from root to tip. The most noticeable feature of the rotor-blade wake in practice is the existence of a strong vortex emanating from the blade tip, where because the velocity is highest the rate of change of lift is greatest. In hover the tip vortex descends below the rotor in a helical path. It can be visualized in a wind tunnel using smoke injection (Fig. 2.12) or other means and is often observable in open flight under conditions of high load and high humidity. An important feature which can be seen in Fig. 2.12 is that on leaving the blade the tip vortex initially moves inwards towards the axis of rotation and stays close under the disc plane: in consequence the next tip to come round receives an upwash, increasing its effective incidence and thereby intensifying the tip vortex strength. Figure 2.13 due to J.P. Jones2 shows a calculated spanwise loading for a Wessex helicopter blade in hover and indicates the tip vortex position on successive passes. The kink in loading distribution at 80% span results from this tip vortex pattern, particularly from the position of the immediately preceding blade. The concentration of the tip vortex can be reduced by design changes such as twisting the tip nose-down, reducing the blade tip,area or special shaping of the planform, but it must be borne in mind that the blade does its best lifting in the tip region where the velocity is high. Since blade loading increases from the root to near the tip (Fig. 2.13), the wake may be expected to contain some inner vorticity in addition to the tip vortex.This might appear as a form of helical sheet akin to that of the illustration in Fig. 2.11, though generally not of uniform strength. Definitive experimental studies by Gray3, Landgrebe4 and their associates have shown this to be the case. Thus the total wake comprises essentially the strong tip vortex and an inner vortex sheet, normally of opposite sign. The situation as established by Gray and Landgrebe is pictured, in a diagram which has become standard, by Bramwell (p. 117) and other authors. Figure 2.14 is a modified version of this diagram, intended to indicate vortex lines making up the inner sheet,emanating from the bound vorticity on the inner part of the blade. The Gray/Landgrebe studies show clearly the contraction of the wake immediately below the rotor disc. Other features which have been observed are that the inner sheet moves downward faster than the tip vortex and that the outer part of the sheet moves faster than the inner part, so the sheet becomes increasingly inclined to the rotor plane.