Frequency calculation from kij and mij

Wind & Water Structural Analysis
1

Dear Jason,

I’m sorry for the many questions i asked you in the last days but maybe this is the last one because I’m going to graduate! I’ ve read your old thesis about FAST_AD and I understood the way “Modes” acts, giving out the Rayleigh-Ritz coefficients.In that thesis you introduced the kij and mij that are function of “shape functions” and allows the calculation of coefficients and frequency . Then I’ve read the “Unofficial Theory Manual” where you introduce the same terms kij and mij…but in this case they are function of the mode shapes and not shape function.In Fast I understood that you use the kij for the calculation of elasric active force of the tower and blades…but you also talk about natural frequancy ( in the case without top mass and gravit. destiffening). So I’d like to know which is the reference theory (similar to that used on FAST_AD with shape functions?) that you use implicitly to caluclate frequencies having the mode shapes!

2

Dear Alessandro,

I’m not sure I fully understand your question. However, the approach taken for the elastic (EI) stiffness matrix is basically the same between the structural module of FAST (called ElastoDyn in FAST v8) and Modes, except that polynomials are used as shape functions in Modes and mode shapes are used as shape functions in ElastoDyn. In both cases, the elastic stiffness matrix, K_ij, is:

K_ij = INTEGRAL( EI(z)*phi_i’‘(z)*phi_j’'(z)*dz, z = 0, L )

where,
z = distance along the beam from the root (z=0) to the tip (z=L)
EI(z) = bending-stiffness distribution along the beam
phi_i’‘(z) = second spatial derivative of the shape function phi_i(z) with respect to z
phi_j’'(z) = second spatial derivative of the shape function phi_j(z) with respect to z

I hope that helps.

Best regards,