Hi,
I'm doing a quasi-static wing flutter analysis.
I annotated the attached worksheet a bit but the basic idea goes like this:
My objective is to find a critical value for k and X as follows:
A, B, D and E are functions of k and X.
They are coefficients to a series of two equations such that:
A*h + B*j = 0
D*h + E*j = 0
where h and j represent another function (not shown in worksheet) - h and j both > 0.
Thus for a solution the determinant of matrix:
must = 0. This is where things get a bit complicated.
The suggested approach here is an excel job of many columns and some linear interpolation (groan):
Set k = some initial guess (2)
Calculate numerical values of those coefficients A, B, D and E in terms of X
Create two equations from the determinant = 0 condition - real and imaginary parts.
Solve to find X which typically has two real solutions and one imaginary (though could also be three imaginary potentially).
(Effectively) repeat for other values of k using a guess-and-correct approach until a critical condition is obtained.
The critical condition I am interested in is when either real solution of X is equal to the imaginary solution.
(Physically, this relates to a condition for the wing where system damping is equal to aerodynamic agitation, the critical point at which wing "flutter" would begin.)
My effort in Mathcad is as follows:
Find the determinant
Find the coefficients of the determinant expressed wrt X
Use polyroots to find the two real and imaginary roots seperately - these come out correct for my initial "guess" of k=2
"Given Real = Imag, Find(x,K)"
Correct values for this case should come out at Xcrit = 1.149 and kcrit = 0.274 (though thiese results invovled some manual linear interpolation - absolutely though 0.27 < kcrit <0.275.
Note that using Minerr gives a k value of 0.266 (which is close but wrong) but doesn't solve at all for X...
