Dear Wystan,

Good question. Iâ€™m surprised this one hasnâ€™t been asked before.

The tower damping in FAST is specified as a fraction of critical damping for each mode of the isolated tower. The damping is implented as a stiffness-proportional modal damping, mathematically as follows. If the iâ€™th generalized elastic force is given as:

Felastic_i = K_ij*q_j,

where K_ij is the (i,j) element of the generalized elastic stiffness matrix of the isolated tower and q_j is the jâ€™th tower-bending degree-of-freedom (DOF), then the iâ€™th generalized damping force is given by:

Fdamping_i = K_ij*beta_j*qdot_j,

where qdot_j is the first time-derivative of q_j and beta_j is the stiffness-proportional constant for the jâ€™th DOF. (Einstein notation is used here.) The stiffness-proportional constant is derived from the modal damping ratio of the jâ€™th tower-bending mode (zeta_j) and the jâ€™th natural frequency of the isolated tower (f_j) as follows:

beta_j = zeta_j/(pi*f_j),

where:

f_j = 1/(2*pi)*SQRT(K_jj/M_jj),

where M_ij is the (i,j) element of the generalized mass matrix of the isolated tower.

A few points of clarification for FAST:

*i and j both go from 1 to 2 for both the fore-aft and side-to-side bending modes.

*zeta_j represents TwrFADmp(j)/100 and TwrSSDmp(j)/100 for the fore-aft and side-to-side bending modes, respectively.

*The generalized elastic stiffness matrix of the tower depends on the specified distributed tower-bending stiffness (including any scaling or tuning factors) and the mode shapes. It does not depend on the gravititional destiffening or the platform or rotor-nacelle assembly (RNA) DOFs, so, it really represents the elastic stiffness only of the isolated tower.

*The generalized mass matrix of the tower depends on the specified distributed tower mass (including any scaling factors) and the mode shapes. It does not depend on the mass of the platform or RNA, so, it really represents the mass only of the isolated tower.

*Because the stiffness-proportional constant is calculated only for the isolated tower, the damping of the tower-bending modes when coupled with the platform and RNA may be different than specified (i.e., the damping ratio derived for the tower-bending modes through a FAST linearization analysis of the full wind turbine system may differ from what is specified for the isolated tower in the FAST input file).

I hope that helps.

Best regards,