Minimum state smoothing method

The matrix An (Фп, i u) has been computed for nK reduced frequencies к1, …,кПк with the assumption of harmonic motion. For arbitrary motions like those defined by Eq. (6), it is necessary to extend the values of the aerody­namic coefficient matrix to an area of the complex plane containing the imagi­nary axis, i. e. to determine An (Фп, p) for p = iu(1 + ia) with a = 0.

The minimum state smoothing method Karpel, 1982 consists in modelling the generalized aerodynamic forces by using a rational approximation and aux­iliary state variables :

V2

K*

gn

0

c*n 0

En – I

M* 0

gn

0 0

An0, An1, An2, Dn and En are matrices with dimensions (mn x mn) for An0, A„i and An 2, (mn x np) for Dn and (np x mn) for En. Rn = diag(ri,…,Гпр) is a diagonal matrix where np is the degree of the denominator of the rational function or the number of poles and r * < 0 are the poles. These matrices are computed by using a method of least squares minimization Poirion, 1995.

Mg-[3] [K*n + Gn (p)] – M;

*—1 c*

gn gn

– ^gn 4" 2 Poo 4(зо Ang, Сgn Cgn + 2 Poo ^ ^oo ^-n 1 >

Mgn = Mgn + ^Poo C[4] A„2 and G„ ip) = v£ {p c/O D„ [ip c/Vqq) I – Rn ]—1En. The matrices Gn (p) and H(p) are complex and depend on V^. The nonlinear eigenvalue problem Eq. (15) is solved for nV increasing velocities V1 ,…,Vnv by using an iterative process based on the method of successive approximations for finding a fixed point of a function Tran et al., 2003.

In Eq. (14), a vector of np auxiliary state variables zn have been defined. They satisfy a system of first-order differential equations in the time domain :

Using Eq. (14), the flitter equation (8) can be written under the form of a non-linear eigenvalue system of dimension 2 mn :

where K*n, C*n and M*n are the same matrices as in Eq. (15). This second – order system is solved using the Newmark numerical integration scheme.