Quasi-steady Aerodynamic Loads on Flexible Rotor Blades

The blade velocity components in the blade fixed undeformed body coordinates at a point x from the blade root are

(4.10.1)

In the above, [uvw] are the elastic translational displacements, vin in the total inflow velocity. If T is the transformation from the undeformed to the deformed coordinates, the blade velocity components in the deformed coordinates are

(4.10.2)

The velocity components to first order in в are

UT = XRx + V,

Up = —XRx @вс + /e + J v’w"dxj + XRkin + w — (hG + /e)V + Qvw’.

In the above, во is the geometric pitch angle including pretwist and controlled pitch input, and /e is the elastic torsion. The local inflow angle is defined as

Ф = tan-1 f. Given, W = Xt

(4.10.5)

(во + Фe)UpUT – Up + (-2^’j U2 – + *ac) UpbRX — (во +

(4.10.6)

c( (вО + фе)ит – UPUt) – (1 + – Lop ^2 + xac^ UTbRX dw (вО + Фе)

(4.10.7)

Linearizing the above expressions,

kin = kf + ki. (4.10.9)

The induced inflow is assumed to be the sum of steady component and a transient component and consequently can be expressed as,

k, = k,0 + Ak,, kin0 = kf + k,0. (4.10.10)

The steady component of the induced inflow is estimated from the steady thrust coefficient as,

Inserting the inflow expression and discarding all higher-order terms,

[LwLvM/ = yPi(2b)(^ X2R2

2P, N /

— Pi(2b)l XR

Lw = уP™(2b) (X2Rxbw’ – X2R2xkin0 – X2R2xDk – X2R2x2(0G + фе))

2p f 3b ■ b Л ’

– у Pm(2b)X^| x>w + (kin0 – 2xhG)V – у x/ + 2XRWJ ’

((4.10.13))

Lv = 2pPi(2b^X2R^k2n0 – x22^ + X2R2(2kin0 – xhG)Dk; – X2R2xkM(hG + /e)j – ^-Pi(2b^X^| (xhG – 2kin0)W + ^2x+ kinQhGjvjj,

(4.10.14)

R

Thrust* = J (Fz – mZ)kdr « J (Lw – mw)kdr. (4.10.15)

Rh Rh

Non-dimensionalizing by pqR4X2 the above relation, the thrust coefficient is

1 1

1 f 1 r m

Ш]Lwdx – xTr! pq“R2wdx – (4Л0Л6)

xh

—— X2R3 I Lwdr = – 2 @ (kin0 + Dki) У brdr + (hG + фе) ^j

1 xh Xh /

1 1 1 f m _ 1 f.. m

X2RJ ppooR2 € Г X2RJ 1W Г’ 1 ppooR2 ’

Hence if,

Ct — 2(^(kin0 + Dki) ^— 2—^ (hG ^ ^ ^—— 3—+ 2 f b2w’rdr

j 0w{rw + (kin0 – 2rhc)v -^Tr/^dr – RJ (l + ^ 0^2 wdr.

(4.10.20)

In steady flow,

Ct0 —-2( + в/1 – rh

In the case of constant inflow with steady geometric pitch, the Pitt-Peters finite state model for the dynamic inflow reduces to,

(4.10.23)