Derivation of the referred parameter relationships

Consider now the most general case of a helicopter in climbing flight at low level. As indicated earlier the power required to maintain a steady flight condition will depend

on:

• weight (W)

[MLT ~2

• forward speed (v)

[LT -1]

• rate of climb (Vc)

[LT -1]

• height (above ground level) (Z)

[L]

• local speed of sound (a)

[LT -1]

• ambient density (p)

[ML -3]

• rotor speed ())

[T -1].

Note that local speed of sound has been included as a means of accounting for compressibility effects on the lift and drag characteristics of the rotor blade. So:

P = f(W, V Vc, Z, a, p, )

Using dimensional analysis yields:

Подпись: P = fw) v к Z) pa!

pa4 ’ a’ a’ a ’ )

Derivation of the referred parameter relationships Подпись: f Подпись: Wm2 v к Zm aa4 ’ a ’ a ’ a

This can be rewritten as:

where m = )/)0 and a = p/p0

Noting that a is a function of ambient temperature:

Pm2 _ J Wm2 V V Zm^

a62Ve _ V a02 ’ V ’ V ’ Ve/

where 0 _ T/T0.

Reorganizing and collecting like terms produces the ‘W/a’ referred power relationship:

p _ J_w V V Z _m_^i

am3 lam2’ mm ’V0)

Since measuring air density is difficult, an alternative grouping can be obtained by replacing a with 8/0:

p _ J w v к Z _®_

8V0 V 8 , m, m, ’V0J

This is called the ‘ W/8’ referred power relationship which, although easier to use since it lacks air density, cannot be used for rotorcraft with fixed rotor speed. Note that in both groupings the forward speed and rate of climb have been expressed as advance ratios.

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