# Changes to FAST Code and New Output Variables

In order to assess the performance of three-bladed turbines with teetering enabled, computer modeling was performed by modifying FAST (Version 8.12), the wind turbine modeling software available from National Renewable Energy Laboratory^{1,2}. The present FAST software code can model turbines with two or three blades. Two bladed turbines can be modeled with teetering either enabled or disabled, whereas three bladed turbines do not permit teetering. The present FAST code enables teetering of two bladed turbines by having a degree of freedom (tDOFya) to balance differences in out-of-plane torque about the Ya axis. The Ya axis, as shown in the following figure, is an axis that rotates with the hub about the main shaft axis, or X axis. The Ya axis is also the axis of the teeter pin and is used by the existing code to enable teetering of blades 1 and 2.

Existing Teetering Rotational Axis Enabled by FAST Code

In adapting this degree of freedom to a hub with three blades, the Ya axis was set perpendicular to the pitch axis of blade 1. The code was further modified by adding a second degree of freedom, tDOFza, to enable blades 2 and 3 to rotate about the Za axis. All three blades can rotate about the Ya axis, however only blades 2 and 3 can rotate about the Za axis since it passes through the pitch axis blade 1. The orientations of Ya and Za axes for three bladed turbines are shown in the following figure.

Rotational Axes added to FAST Code

The out-of-plane rotations of blade 1 (about the Ya axis) are designated as follows using the existing FAST software.

TeetPya: Rotor teeter angular position of blade 1.

TeetVya: Rotor teeter angular velocity of blade 1.

TeetAya: Rotor teeter angular acceleration of blade 1.

In the case of three-bladed teetering, these designations can also be used to describe the rotations of the hub about the Ya axis as follows:

TeetPya: Rotational position of hub about Ya axis.

TeetVya: Rotational velocity of hub about Ya axis.

TeetAya: Rotational acceleration of hub about Ya axis.

These rotations describe the teetering of blade 1 versus the combination of blades 2 and 3. With three-bladed teetering, the following new output parameters (in bold) are introduced to describe the out-of-plane rotations of the hub about the Za axis:

**TeetPza**: Rotational position of hub about Za axis.

**TeetVza**: Rotational velocity of hub about Za axis.

**TeetAza**: Rotational acceleration of hub about Za axis.

These rotations describe the teetering of blade 2 vs blade 3. Blade 1 is not involved because the Za axis is collinear with the pitch axis of blade 1. The modified FAST software determines the values of the rotational positions, velocities and accelerations about the Ya and Za axes. From these calculated values, the following new output variables are introduced to described the teetering motions of blades 2 and 3.

**TeetP2a** = Rotor teeter angular position of blade 2 = sin (210°) * TeetPya - cos (210°) * TeetPza

**TeetV2a** = Rotor teeter angular velocity of blade 2 = sin (210°) * TeetVya - cos (210°) * TeetVza

**TeetA2a** = Rotor teeter angular acceleration of blade 2 = sin (210°) * TeetAya - cos (210°) * TeetAza

**TeetP3a** = Rotor teeter angular position of blade 3 = sin (330°) * TeetPya - cos (330°) * TeetPza

**TeetV3a** = Rotor teeter angular velocity of blade 3 = sin (330°) * TeetVya - cos (330°) * TeetVza

**TeetA3a** = Rotor teeter angular acceleration of blade 3 = sin (330°) * TeetAya - cos (330°) * TeetAza

where blades 1, 2 and 3 are positioned at 90°, 210° and 330°, respectively when viewed from the Nacelle. These equations apply when blades rotate clockwise as viewed from the front or counter-clockwise as viewed from the Nacelle. Summing the above equations for the teeter deflections of blades 1, 2 and 3 gives a value of zero. Although three blades teeter, there are only two degrees of freedom since subtracting the teetering angles, velocities, or accelerations of any two blades from zero determines the teetering angle, velocity or acceleration of the remaining blade.

Additional output parameters have been introduced in order to describe teetering motions about stationary Ys and Zs axes as shown below.

The stationary ys and zs are both imaginary, rather than physical axes. These parameters are also determined from the rotations about the rotating Ya and Za axes as follows:

**HubRotPys** = Hub rotation angle (position) about fixed ys axis = TeetPya * sin (Azimuth) + TeetPza * cos (Azimuth)
**HubRotPzs** = Hub rotation angle (position) about fixed zs axis = -TeetPza * cos (Azimuth) + TeetPza * sin (Azimuth)
**HubRotVys** = Hub rotation velocity about fixed ys axis = TeetVya * sin (Azimuth) + TeetVza * cos (Azimuth)
**HubRotVzs** = Hub rotation velocity about fixed zs axis = -TeetVza * cos (Azimuth) + TeetVza * sin (Azimuth)
**HubRotAys** = Hub rotation acceleration about fixed ys axis = TeetAya * sin (Azimuth) + TeetAza * cos (Azimuth) **HubRotAzs** = Hub rotation acceleration about fixed zs axis = -TeetAza * cos (Azimuth) + TeetAza * sin (Azimuth)

where the Azimuth is the angle of blade 1. Modeling data include graphical output of these rotations. These Ys and Zs rotations provide a unique feature of a three bladed teetering hub in that the rotor axis can differ from the main shaft axis. An example of this is shown in Figure 2.

[1] Jonkman, J. M.; Buhl Jr., M. L. "FAST User's Guide," NREL/EL-500-29798. Golden, Colorado: National Renewable Energy Laboratory, 2005.

[2] NWTC Computer-Aided Engineering Tools (FAST by Jason Jonkman, Ph.D.). http://wind.nrel.gov/designcodes/simulators/fast/. Last modified 27-February-2013; accessed 2-June-2013