Dear all,

I am studying the behavior of the teeter degree of freedom as I came across the helicopter flapping equation in Burton’s Wind Energy Handbook. In short, the equation for teeter angle `β`

is

```
Iyy (d^2𝛽)/(dt^2)+ Ca (d𝛽/dt) + (Ka + Kcf)𝛽 = Mext
```

where `Iyy=mass moment of inertia around teeter axis, Ca = aerodynamic damping, Ka = aerodynamic stiffness, Kcf = centrifugal stiffness, Mext = Forcing moment (usually due to aerodynamic imbalance).`

Now, I know from the docs that FAST implements this feature uses Kane’s equations which considers an inertial frame of reference instead of rotational. Moreover, `Ca, Ka`

are not really constant coefficients, but are determined using AeroDyn in each timestep.

Nevertheless, for moderate to high tip speed ratios and for the condition that there is no stall, one can estimate these coefficients using the table in Schorbach et al. ( DOI: 10.1002/we.1823).

I decided to put this to the test by simulating the UAE phase VI turbine from the r-tests with and without the teeter degree of freedom. Tilt was set to 30 degrees and the rotor spun at `TSR = 6, U = 3.77m/s`

. Due to the tilt conditions I used OLAF. Using the data in report NREL/TP-500-29955 the aerodyanamic damping coefficient and rotor inertia are estimated. Note that `Ka`

equals zero as `d3 = 0`

.

The `2nd Order ODE`

is implemented in Python where `Ca=3100Nm/rad, RPM = 5.2rad/s, Iyy=945kgm^2`

Moreover, `M_ext`

is determined by the difference between the AeroDyn Output variables B1AeroMy and B2AeroMy for the rotor that has no teeter degree of freedom. These are plotted below:

Thus, the moment imbalance between the blades that gives `Mext = M0*sin(Omega*t)`

, with $M0=53Nm$. Note, this is determined using the non-teetered rotor! This is the forcing function that can be used in the `ODE`

where the `ODE`

is describing the teetered rotor. As a sanity check, I derived the function for the flapping angle myself also using Kane’s equation. Thus, the first method that uses the aforementioned `ODE`

is in the rotational reference frame which introduces the centrifugal stiffening. My own derivation using Kane’s equations is in an inertial frame thus not centrifugal term appear.

Both methods (ODE, Kane’s) are compared with the outcome of OpenFAST/ElastoDyn (also Kane’s) as plotted below.

Clearly, ODE and Kane’s have a similar magnitude whereas ElastoDyn differs with a factor 1.5. In terms of absolute angle this might seem insignificant, but the reaction forces in the system will be significantly higher for higher teeter angle (this implies higher teeter acceleration). Note that there is a phase difference between ODE, Kane’s on the one hand and OpenFAST on the other hand. This is because the sinusoidal function should contain a phase shift, but for this discussion it is less relevant.

Based on these observations I have the following questions:

- Has OpenFAST/ElastoDyn teeter D.O.F. been verified/validated? If so, are there some references?
- Why is
`B1AeroMy-B2AeroMy`

slightly smaller than`RtAeroMyh`

? - Is the assumption that the moment imbalance for the non-teetered rotor is the forcing moment for the teetered rotor correct? (This has been assumed in the example provided in Burton’s.)

BR,

Luc