Proposed Resonance in Two-Bladed Teetering

Dr. Andrew Garrad published several papers [1,2] on the forced dynamic response of wind-turbine rotors. The reasoning is that basically, a teetering turbine has a natural frequency at 1P (1/rev) which is the same natural frequency as wind shear and tower shadow. Because of this, the teetering motion of such a rotor is fundamentally resonant. Garrad also showed that a teetering rotor is fundamentally unstable unless its natural frequency can be shifted substantially away from 1P, or adequate damping can be maintained. Because of this, manufacturers of two-bladed, teetering wind turbines have adapted strategies of using very tight limits on teetering and pitch-teeter coupling to mitigate the effects of resonant teetering. One manufacturer imposes teetering limits of ± 2.2°.

Why Resonance is Unlikely with Three Bladed Teetering Hubs

No resonance has been observed with all modeling of three-bladed, teetering wind turbines as long as teetering does not lead to a change in pitch angle.  Furthermore, it is proposed that mechanical resonant teetering likely cannot occur with a three-bladed teetering hub for the following reasons.

1. With mechanical resonance, an external force drives another system to oscillate with greater amplitude at specific frequencies.  An example would be a parent pushing a child on a swing in time and causing the child to go higher and higher with each push.  Another example would be the collapse of the Broughton Suspension Bridge where the rhythmic force created by the soldiers marching in cadence occurred at a natural frequency of the bridge, causing a resonance with and eventual collapse of the bridge.  With three-bladed teetering, the rotor cannot apply an out-of-plane bending torque to the main shaft.  This is shown with modeling where values of LSSTipMya and LSSTipMza are zero as long as teetering stops are not reached.  This is also shown with a working model.

2. The teetering profile is determined by the wind profile that includes tower shadow, wind shear and turbulence.  As shown in Yaw Mechanism, rotation of the wind turbine about the yaw axis causes a temporary unstable teetering profile.  The wind forces rapidly re-establish a stable teetering profile by rotating the rotor about the vertical Zs axis.  Consequently, if teetering were to cause a resonance with another part of the wind turbine and that resonance were to change the teetering profile, the change in the teetering profile would likely be overcome by wind forces seeking to re-establish a stable teetering profile.

3. Teetering with three blades would disrupt the possible onset of resonant teetering because the added blade and the added teetering degree of freedom add measures of randomness. 

Why Resonance is More Likely with a Three Bladed Rigid Hub

It is proposed that a three-bladed, teetering hub would actually be less likely to experience resonance than would a three-bladed rigid hub.  This is based upon the fact that a rigid hub can transfer out-of-plane torque to the main shaft (LSSTipMya and LSSTipMza), whereas a three-bladed teetering hub cannot.  For example, it would be possible for a rigid hub to resonate with other parts of the wind turbine at 4P by means of this torque transfer, whereas it would not be possible for a teetering hub.   A possible example of this is presented in a paper by Fleming et al. where a conversion of a two-bladed teetering hub to a three-bladed rigid hub resulted in "an unstable 2.7Hz drivetrain oscillation at rated speed".  The paper indicated that this resonance occurred at 4P and did provide possible causes for these vibrations, however it is possible that the transfer from a teetering hub to a rigid hub played a role since two-bladed teetering also reduces the out-of-plane bending moments applied to the shaft tip.  A 4P resonance (2.7Hz) is possible because it is a harmonic of 1P.  It would be the equivalent of a parent pushing the child every fourth swing.  If this resonance is due to out-of-plane torque applied to the main shaft at 1P, it would be expected that the oscillation at 2.7Hz would be observed with the rigid hub and not with the teetering hub.   View paper » 

1. Dynamics of Wind Turbines, AD Garrad, Proc. IEEE, Vol 130 Issue 9 (1983) and The Prediction of Teeter Excursions on a Horizontal Axis Wind Turbine, AD Garrad, Proc. 4th BWEA Conference, Cranfield, UK (1982)
2. The Analysis and Design Implications of Pitch-Teeter Coupling, GM Henderson, RS Haines and DC Quarton, Proc 11th BWEA Conference, Glasgow (1989)