Solution of the coupled system. Double scanning method

For motions defined by Eq. (6) in the frequency domain, the aerodynamic coefficient matrix An (Фп, p) defined in Eq. (11) depends only on the quotient pc/V^ and can be written as An (Фп, pc/Vrx) where c is a reference length, for exemple the blade chord. Let’s define the reduced frequency к = u c/V^.

Using Eq. (11) and denoting An(Фп, pc/Vrx) and A^(Фп, pc/V! X>) the real and imaginary parts of A n (Фп, pc/Vrx:), the flitter equation (8) can be written

under the form of a non-linear eigenvalue system of dimension 2 m n :

orH(pc/V^) x = px, with K*„(pc/F00)=KS„ + y<X)V20k’n{&n, pclV00) and C*„ (p с/) = Cs„ + cp^ V^A" (Ф„, p с/) / (p с/).

If the damping factor is small (|a| ^ 1), we have pc/Vn ~ i к and the matrices A(Фп, pc/Vn), An(Фп, pc/Vn) and H(pc/Vn) can be approxi­mated by An (Фп, iк), An (Фп, iк) and H(i к). Eq. (12) becomes :

or H(i/c)x = px, with K*„(i/t) = Ks„ + ^Роо^А^Фп. ік), C*n(i/c) = cs„ + і cPqq А"(Ф„, in)/к and к = Srm(p) c/V^. H(i/t) is a real ma­trix which depends on An(Фп, iк), An(Фп, iк) and Vn. The solutions of the approximated eigensystem (13) should be determined for nV increasing velocities V^ ,…,Vcny and the frequencies should satisfy u = 3m(p) = кVcю/c. For each velocity V^, the following eigensytems are solved for i = 1,…, 2 mn and j = 0,1,2,…, until the convergence on к is obtained:

H(i «i, j ) xi, j + 1 pi, j +1 Xi, j + 1, кІ, І U*,j c/Vrx ^m(pi, j ) c/Vn,

where (p*-, xj j) are the i-th complex eigenvalue and eigenvector obtained at the j-th iteration. The matrix H(i к*-) is computed by interpolating A, n (Фп, i к) and An (Фп, i к)/к from the tabulated values. It is remarked that the double scanning method is not valid for small velocities since the latter lead to very large reduced frequencies and the extrapolation of H(i к*-) will be incorrect.