Energy2green Wind And Solar Power System

The loadings on the hub consist of the aerodynamic, gravity and inertia loadings on the blades and the equal and opposite (discounting hub self-weight) reaction from the shaft. For fixed hub machines, the loading on the shaft will include a significant moment arising from blade aerodynamic loads, but in the case of teetered two bladed rotors this moment will be virtually eliminated. In either case, however, the cantilevered low-speed shaft will experience large fluctuating moments due to rotor weight as it rotates. Figure 5.39 shows a low-speed shaft and front bearing in a factory prior to assembly.

The shaft moments due to out-of-plane loads on the blades can be expressed as moments about a pair of rotating axes, one perpendicular to blade 1 and the other parallel to it. In the case of a three-bladed rotor, these moments are respectively as follows:

Mys = AMyi - 2(AMY2 + AMY3) MZS = ^-(AMY3 - AMyi) (5.118)

Here AMY1, AMY2 and AMY- are the fluctuations of the blade out-of-plane moments about the hub centre (MY1, MY2 and MY-) about the mean value (see Figure 5.40).

The deterministic aerodynamic loads on the rotor may be split up into a steady component, equal for each blade, and a periodic component, also equal for each

MY2I

Blade 3

Figure 5.40 Shaft Bending Moments, with Rotating Axis System Referred to Blade 1

blade, but with differing phase angles. The blade root out-of-plane bending moments due to the first component will be in equilibrium, and will apply a 'dishing' moment to the hub which will result in tensile stresses in the front and compression stresses in the rear. These stresses will be uniaxial for a two bladed rotor, and biaxial for a three-bladed rotor.

The fluctuations in out-of-plane blade root bending moment due to wind shear, shaft tilt and yaw misalignment will often be approximately sinusoidal, with a frequency equal to the rotational frequency. Using Equations (5.118), it is easily shown that, for a sinusoidally varying blade root bending moment with amplitude Mo, the amplitude of the resulting shaft bending moment is 1.5Mo for a three-bladed machine and 2Mo for a rigid hub two-bladed machine.

In the case of wind shear conforming to a power law, the loading on a horizontal blade is always greater than the average of the loadings on blades pointing vertically upwards and downwards, so the loading departs significantly from sinusoidal. The shaft bending moment fluctuations due to wind shear with a 0.2

exponent are compared in Figure 5.41 for two- and three-bladed rigid hub machines operating at 30 r.p.m. in a hub-height wind speed of 12 m/s. The ratio of moment ranges is still close to 2:1.5.

The out-of-plane blade root bending moments arising from stochastic loads on the rotor will result in both a fluctuating hub 'dishing' moment (see above) and fluctuating shaft bending moments. For a two-bladed, rigid hub rotor, the shaft moment is equal to the difference between the two out-of-plane blade root bending moments, or teeter moment, the standard deviation of which is given by Equation (5.102). Similarly, the standard deviation of the mean of these two moments (i.e., the 'dishing' moment) is given by Equation (5.103).

The derivation of the standard deviation of the shaft moment for a three-bladed machine is at first sight more complicated, as the integration has to be carried out over three blades instead of two. However, if the shaft moment about an axis parallel to one of the blades, Mzs (Figure 5.35), is chosen, the contribution of loading on that blade disappears, and the expression for the shaft moment standard deviation becomes:

Mzs iAQ

R v3 v3

where the limits of the integrations refer to the other two blades. kou(r1, r2, 0) is given by Equation (5.51), with Qr set equal to zero when r1 and r2 are radii to points on the same blade, and replaced by 2^/3 when r1 and r2 are radii to points on different blades. Note that, compared with the two-bladed case, the cross correla-

Shaft bending moment: two bladed rigid hub machine

Shaft bending moment: three bladed machine

Rotor diameter = 40 m Rotational speed = 30 r.p.m. Hub-height wind speed = 12 m/s Shear exponent = 0.2

Rotor diameter = 40 m Rotational speed = 30 r.p.m. Hub-height wind speed = 12 m/s Shear exponent = 0.2

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