# Verification of teeter dynamics

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

Dear @Luc.vanBeek,

Here are my responses to your questions:

Regarding verification/validation, I know that there was some verification/validation of teeter many years ago with much older versions of FAST, but nothing recently. I can’t think of a good old reference to share with you.

Regarding `B1AeroMy-B2AeroMy` being slightly different from `RtAeroMyh`, I would guess this would have to do with the the different coordinate systems these two outputs are reported in. The former are reported in the blade root coordinate system while the later are related in the hub coordinate.

Regarding the moment balance, I would guess the first-order term could come from the fixed teeter condition, but there are likely aerodynamic interactions associated with teeter that would introduce higher-order effects.

Best regards,

Dear @Jason.Jonkman,

Thank you for you response;

• It’s a pity that you do not have any references. But maybe you could point me to some databases. For example, is there an NREL or even NASA database where it can be found? That being said, I found a paper by Malcolm (2000) “How a teetered rotor with delta-3 really works” and he relies on ADAMS. I can derive some verification data from that paper.

• Ok, that explains the differences.

• How would that work? In essence, the damping due to teeter motion is captured in the damping term. I could imagine some additional flow conditions such as the location of the tip vortices with respect to the blade leading to a different moment. However, when considering the `ODE`, in order have a magnitude difference of factor `1.5` for the teeter angle, the forcing moment should be a factor `1.5` higher (Think of the analytical solution for a mass-spring-damper system). I can hardly imagine any higher-order effects giving rise to such a large moment. Even my suggestion of the location of the tip-vortices is a far-fetched reason because the final solution is in the order of `0.3 degrees`. Could you elaborate on other aerodynamic interactions?

Best regards,
Luc

Dear @Luc.vanBeek,

I would say that in your teeter equation, Ca is the linear change in aerodynamic moment with teeter rate, Ka is the linear change in aerodynamic moment with teeter angle, and you said Mext is the aerodynamic moment in the absence of teeter. However, there could be higher order-effects not accounted for in your equation such as the quadratic dependencies of the aerodynamic moment associated with teeter angle and teeter rate, which would be understandable because aerodynamic forces at the element level are proportional to the square of the relative velocity. Rotor aero-elasticity not accounted for your equation could also impact the teeter response.

Best regards,