Multiblade Coordinates

The aerodynamically coupled flap-pitch equations of motion of a single blade have been derived in a rotating frame as function of the azimuth angle W, When W = Wt, the equations represent the motion of the tth blade. To derive the equations of motion of all the blades as a single unit, the coefficients in the equations may be expressed in terms of the so-called multiblade coordinates. This is done by
expanding all trigonometric functions such as products of sine and cosine func­tions as the sums of relevant sine and cosine terms. Groups of terms containing factors 1, cos(W), sin(W), … may be identified and individually set equal to zero to give N equations of motion for an N-bladed rotor for the fixed frame harmonics of each blade. Thus, the fixed frame equations of motion obtained by applying multiblade coordinate transformations will represent the dynamics of the rotor disc containing N blades. For example, if the flap modal coordinates in the kth mode for an N-bladed rotor are bk1, bk2, bk3, ■ ■■, bm, they are transformed by discrete Fourier series transformations to fixed frame coefficients, bk0, bk1c, bk1s, ■ ■ ■; bkd, by,

1 N 2 N 2 N

bk0 = N^2 bkm; bknc = N^2 bkmCOs(nWm); bkns = nE bkm sin(nWm);

m=1 m=1 m=1

1N

bkd = bkm (-1)m,

m=1

(4.7.21)

where the last transformation exists only for even numbers of blades. For a four – bladed rotor, one has bk0, bk1c, bk1s, bkd, while for a three-bladed rotor, one has bk0, bk1c, bk1s. As most wind turbine rotors are three-bladed, we shall assume that the number of blades is 3. Moreover, the above four transformations will be referred to as the collective, cosine-cyclic, sine-cyclic, and differential collective operators.

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