# Analysis of a climbing helicopter using momentum theory

Before describing how the experimental method can be adapted for use in climb performance testing it is worthwhile to use simple momentum theory to analyse the factors affecting a helicopter in climbing flight. Figure 3.24 shows the change in velocity that occurs as flow passes through a climbing actuator disk.

As before, analysis begins by balancing the forces acting on the centre of gravity:

T cos yd = mg + d, = mg +1 p V 2 Sv T sin yd = df = 2 pV 2 Sf Fig. 3.24 Momentum disk theory applied to a climbing rotor.

Therefore:

pv 2 sf

tan yd = ——————- ■—f—

2mg + pV 2 Sv

Note how the airspeed of the helicopter has been broken down into horizontal and vertical components, each responsible for drag generation (calculated using the concept of drag area). Applying the elliptic wing analogy:

T = 2pAvV’

Now:   V’ = V Vf + vi sin yd )2 + (Vv + vi cos yd )2

Equation (3.1) can be solved iteratively to yield the variation of induced velocity with forward speed and vertical velocity, see Fig. 3.25. From the figure it can be seen that for typical forward velocities (в 3 vih) the variation of induced velocity with ROC is negligible. This is a very convenient result since it means that the extra power required to climb can be simply added to that required for level flight. Thus the climb power is given by:

Pclimb = 1.2Tvi +1 pV 3 Sf + 8 pbcRCDVT (1 + 4.3p2) + TVv +1 pVv3 Sv (3.2) Fig. 3.25 Variation of induced velocity with RoC and TAS.

Alternatively if the power available, in excess of the power required for level flight (PFLF), is known then:

1 „

P = TV + – nV3 S

x excess v 1 2 г V ^V

and if the parasitic drag and download are small compared with all-up-weight then:

P

V = .excess (3.3)

mg

Equation (3.3) suggests that the climb performance is directly proportional to the excess power available and therefore the forward speed for maximum ROC (VY) will equate to the minimum power speed (VMP).