OpenFast 2nd order Linearization

Dear David,

The linearization of FAST v7 was limited in its ability to linearize anything other than the structural equations of motion. OpenFAST linearization also supports linearization of the hydrodynamics, aerodynamics, etc., which have equations that do not involve second time derivatives. So, the intent is for the OpenFAST linearization to focus on the generation of first-order systems (with equations that involve second time derivatives converted to first order states).

That said, the mass matrix for the structural system can often be inferred from other matrices generated through the linearization process. E.g., for the six platform degrees of freedom of ElastoDyn, which has inputs (u) that are loads (forces/moments, F) and outputs (y) that are accelerations (a), it is understood that:

``F = M*a``

or equivalently

``a = M^-1*F``

and from the linearization process

``dy = D*du``

implies that

``M^-1 = D``

or

``M = D^-1``

That is, the 6x6 rigid-body mass matrix (M) of ElastoDyn can be found by taking the matrix inverse of the 6x6 subset of the input-transmission matrix (D), associated with the 6 platform load inputs and 6 platform acceleration outputs.

And once you have M, you can derive the stiffness (K) and damping (C) matrices from the state matrix (A) because A often has the form;

``A = [ 0, I; -M^-1*K, -M^-1*C ]``

I hope that helps.

Best regards,