2variable Given... Find involving complex matrix determinant

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...