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Substantial differentiation matrix

Substantial differentiation matrix

Substantial differentiation matrix

(OP)
Hello,

I am trying to understand better what the Nastran solutions are actually doing in the background, and upon studying the aeroelastic solution I learned that the matrix of aerodynamic influence is formed by:

[Qkk] = [Skj] * [Ajj]^(-1) * [D1JK + i*k*D2JK]

While I understand the integration and the aerodynamic influence coefficients matrices, I don't understand the so called "substantial differentiation matrices". I tried outputting then to get a grasp of how they are formed, but it remains obscure to me. While reading the references in the MSC.Nastran manual, I found the formula for the downwash in terms of modal displacements, which is given by:

wi/U∞ = dfi/d(x/s) + i*k*(2*s/c)*fi

Which I suppose is the formula being used for the substantial differentiation matrices. However, I still don't fully comprehend what is being done here. Where does this fi come from? Is the s and c referencing the whole wing or the element? I tried calculating this and comparing to what Nastran is generating and couldn't get to the same answers.

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