XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX''"> Aeroelastic Matrices
MSC.Nastran analysis is based on placing degrees of freedom in sets and, for the purposes of this discussion, the aeroelastic equation can be developed in the f‑set. This is the set of equations for the structural model that remains after single and multipoint constraints have been removed. The aerodynamics are splined to these points to produce an overall equation of the form.
(E‑1)
The f-set has been divided into an "analysis" set indicated by the "a" subscript and an omit set with the "o" subscript. The M, K, Q, and P matrices refer to Mass, Stiffness, aerodynamic, and applied loads, respectively while the u terms refer to displacements. The ( ŽŽ ) superscript indicates acceleration and the x subscript on u refers to the aerodynamic extra-points that have been introduced in
(A‑2). It should be noted that in FlightLoads, the first term of the ux vector is the intercept; that is, in addition to the control surfaces and rigid body motions of
Analysis, there is a term that represents zero angle of attack forces that could be due to user input twist and/or can be due to user input pressure forces on the aerodynamic elements.
There are two aerodynamic matrices in
(E‑1), and these are derived from splining the two aerodynamic matrices of
(A‑4) and
(E‑5) to the f-set degrees of freedom:
(E‑2)
(E‑3)
Qff is the aerodynamic correction matrix in that it provides the forces that are produced by the structural deflections. Qfx is the aerodynamic load matrix and provides the distributed load increment for each of the aerodynamic extra-points.
The spline matrices of
(E‑2) and
(E‑3) contain a p superscript to denote the force transform or a d superscript to denote a force transform.
As a final note on
(E‑1), it is noted that splining to omitted (o-set) degrees of freedom is not allowed so that Qao = Qoa = Qoo = Qox = 0.
(E‑1) can now be reduced to the a-set to give:
(E‑4)
(E‑5)
(E‑6)
(E‑7)
(E‑8)
where x can be either "p" or "d" for the force or displacement spline, respectively.
(E‑9)
(E‑10)
(E‑11)
The structural displacements are recovered using standard recovery procedures; noting that
(E‑12)
In the typical case for which there are free structural accelerations, divides the a-set into a r-set and an l-set. The r-set (reference set) contains user defined degrees of freedom equal in number to the number of rigid body motions permitted for the vehicle while the l-set (leftover set) contains the remaining degrees of freedom. There is a relationship between the aerodynamic extra points and the accelerations in the r-set that can be written as:
(E‑13)
where the subscript R denotes the aerodynamic reference coordinate system and the
matrix is a Boolean matrix that selects the aerodynamic reference point accelerations from the vector of trim parameters.
Under quasi-static analysis, the total free accelerations can be related to the rigid body accelerations as:
(E‑14)
where the
matrix is a transformation based on the geometry of the structural model, but which can be derived from partitions of the stiffness matrix:
(E‑15)
