Dear Matthias,

Bending is related to curvature and takes place about the local structural principle axes about which the bending mode shapes are defined. The coupling between flap and edge results by trigonometrically rotating (through the structural pretwist) the flap and edge curvatures into in-plane and out-of-plane curvatures, then integrating twice to get displacement. It is true that if the structural pretwist is nonzero that you should see flapwise and edgewise frequencies in all tip displacements. As I said in my prior post, however, because the blade flapwise stiffness is typically quite a bit less than the edgwise stiffness, the influence of the edgwise bending on the flapwise tip deflection (TipDxbi) will be much less than the influence of flapwise bending on the edgewise tip deflection (TipDybi).

Pages 3-4 of the following paper describe the basic concept behind how FAST couples flapwise and edgewise deflections with axial deflections: nrel.gov/docs/fy04osti/35077.pdf.

Unfortunately, we’ve never been budged the time to write an official FAST theory manual. Nevertheless, here is a quick overview of the FAST theory basis, plus links to where you can find more information.

FAST is a nonlinear time-domain model, which solves equations of motion that are of the form:

M(q,u,t)*qdd = f(q,qd,u,ud,t)

where,

M = mass matrix depending on a nonlinear combination of displacements (q), control inputs (u), and time (t)

qdd = accelerations

f = forcing vector depending on a nonlinear combination of displacements (q), velocities (qd), control inputs (u), wind inputs (ud) (input disturbances), and time (t)

The FAST model is a combined modal and multibody dynamics formulation. The equations of motion above represent the standard multibody dynamics form (with no constraints). The modal part comes in through how the DOFs for the flexible tower and blades are defined. The tower and blades in FAST are treated as elements whose flexibility is determined by the summation of shape functions (modes) scaled in magnitude (DOFs).

I’ve provided a few references below which expand on this information:

*Chapters 2-4 of my Master’s thesis provide a good overview of the basic aerodynamic and structural dynamic theories, although not all of the information is up-to-date (it is about 8 years old), it doesn’t cover everything, and there a few errors: nrel.gov/docs/fy04osti/34755.pdf.

*This paper provides a basic, 10-pg overview of the FAST code and the changes we made to it about 6-7 years ago: nrel.gov/docs/fy04osti/35077.pdf.

*The aerodynamic theory is also described in AeroDyn’s theory manual: nrel.gov/docs/fy05osti/36881.pdf.

*The new theory pertaining to hydrodynamics and mooring system responses is described here: www3.interscience.wiley.com/cgi- … 5/PDFSTART.

Although the above references are not entirely up-to-date, I think they should provide the information you are seeking. However, if you still want more information, such as the detailed derivation of the equations of motion as currently implemented, we do have this information available. However, it is written down in a series very large MS Word documents containing mostly equations and very little explanation. We hope to eventually publish this information as the official FAST theory manual (if we’re ever budgeted the time). But if you need this information as well, we can provide this too.

Best regards,