## Constraints on firstmode natural frequency

As noted in Section 6.14, it is important to avoid the excitation of resonant tower oscillations by rotor thrust fluctuations at blade-passing frequency or, to a lesser extent, at rotational frequency. Dynamic magnification impacts directly on fatigue loads, so the further the first-mode tower natural frequency is from the exciting frequencies, the better. Unfortunately, it is generally the case that the natural frequency of a tower designed to be of adequate strength for extreme loads is of the same order of magnitude as the blade-passing frequency.

In the case of machines operating at one of two fixed speeds, the latitude available for the selection of the tower natural frequency is more restricted. Figure 7.38 shows the variation of dynamic magnification factor with tower natural frequency for excitation at upper and lower blade-passing and rotational frequencies for a three-bladed machine with a 3:2 ratio between the upper and lower speeds. The curves are plotted for a damping ratio of zero, but the difference if the curves were plotted for a realistic damping ratio of about 2 percent would be imperceptible. The figure also shows the tower natural frequency bands available if the dynamic magnification ratio were to be limited to 4 for all four sources of excitation. It is apparent that the minimum dynamic magnification ratio obtainable with a tower natural frequency between the upper and lower blade-passing frequencies is 2.6, for a tower natural frequency of 0.85 times the upper blade-passing frequency. However, in view of the fact that the rotor thrust load

Tower natural frequency/Upper blade-passing frequency

Figure 7.38 Variation of Dynamic Magnification Factor with Tower Natural Frequency for a Two-speed, Three-bladed Machine

Tower natural frequency/Upper blade-passing frequency

Figure 7.38 Variation of Dynamic Magnification Factor with Tower Natural Frequency for a Two-speed, Three-bladed Machine fluctuations will be significantly smaller during operation at the lower rotational speed, it would be advisable to select a somewhat lower tower natural frequency than this to minimize overall fatigue damage.

Once a satisfactory tower design - in terms of strength and natural frequency-has been evolved for a given turbine, it is a straightforward matter to scale up the machine to larger rotor sizes, provided all the tower dimensions are scaled similarly, the hub-height wind speed is unchanged, and the tip speed is maintained constant. It can be shown that in these circumstances the tower natural frequency varies inversely with rotor diameter, as does the rotational speed of the rotor, so that the dynamic magnification factors are unchanged. Similarly, tower stresses due to extreme wind loading are the same as before.

The situation is less straightforward if the tower height is to be varied for a particular turbine. Assuming, as before, that the extreme hub-height wind speed remains the same, and that the wind loading on the tower is negligible compared with the wind loading on the rotor, then the tower base overturning moment is simply proportional to hub height H. Constant stresses can be maintained at the tower base by scaling all cross section dimensions up in proportion to the cube root of the hub height. If the same scaling is maintained all the way up the tower, then the tower natural frequency will vary as \JIB/ H3 = \J H4/3/ H3 = 1/ H5/6, neglecting tower mass, where IB is the second moment of area of the tower base cross section. Thus doubling the tower height would result in a 44 percent reduction in natural frequency. Alternatively, if the tower base overturning moment were assumed to vary as H1 5 to allow for the effect of wind shear on hub-height wind speed and the contribution of wind loading on the tower, then constant tower base stresses could be maintained by scaling the cross section dimensions up by y/H. On this basis, tower natural frequency would vary as 1/\[H. The practical consequences of 'tuning' the tower natural frequency are discussed with respect to tubular towers in the next section.

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