Reduced coupled system

A structure with cyclic symmetry is composed of N identical sectors S0, Si,…, SN – i which close up on themselves to form a circular system. The whole structure is obtained by N — 1 repeated rotations of a reference sector S0 through the angle в = 2 n/N. Each sector is limited by a left and a right frontier L and Lr with the adjacent sectors. The fliid surrounding the structure is also assumed to have the same cyclic symmetry while the applied external forces can vary arbitrarily from one sector to another sector.

Using the cyclic symmetry properties, the motion equation of the structure comes down to N motion equations of the reference sector S0, in terms of the travelling wave coordinates and with the appropriate second members and boundary conditions. Only sector S0 has then to be modelized and, after a finite element discretization, the following reduced matrix systems will be solved to obtain the vector of the travelling wave coordinates un = un (S0,t) of sector S0, for each phase number n = 0,…,N — 1 and with an = пв:

Kun + Clin + Miin = fan (un, un) + fn + rn, (1)

1 N—1 . N—1

fan = ^ E fa(^)e_ifc<Jre and = ^ E (2)

k=0 k=0

K is the stiffness matrix of sector S0, including the centrifugal stress stiffening and the spin softening effects, C is the damping and gyroscopic effect matrix and M is the mass matrix. fa (Sk) is the vector of the aerodynamic forces
applied to sector Sk and which depends on the displacements and the velocities of sector Sk. f(Sk) is the vector of the other external forces applied to sector Sk, including the centrifugal forces. rn is the vector of the interface reactions applied to the frontiers of So with the adjacent sectors, it does not intervene in the solutions of Eqs. (1-3) and it is only present due to the contraints Eq. (3). The cyclic symmetry boundary conditions Eq. (3) are expressed in the cylindrical coordinate system. It is remarked that, the aerodynamic forces fan applied to the coordinates un in Eq. (1) depend only on un and not on the other travelling wave coordinates um for m = n, and they are equal to the physical aerodynamic forces fa (S0, un, Un) induced by un on sector S0.

For each phase number n, the travelling wave coordinates are expressed as a linear combination of the first mn complex modes Фп of the undamped, rotating structure in vacuum (solutions of Eqs. (1-3) without C, fn and fan):

un = фп qn• (4)

Introducing Eq. (4) in Eq. (1) and premultiplying by 4Ф„, we obtain the reduced coupled system in the mn complex modal coordinates qn (t):

Kgn qn + Cgn qn + Mgn fin fagn (Фп qn, Фп qn) + fgn, (5)

un(t) = un ept and qn(t) = qn ept with p = iш (1 + ia), (6)

where ш is the unknown aeroelastic eigenfrequency (ш > 0) and a is the un­known aeroelastic damping factor (a Є ). Using the hypothesis of linearity, the generalized aerodynamic forces are written as :

fagn (Фп qn, Фп qn) Fagn (Фп ,p) qn e • (7)

Substituting Eqs. (6) and (7) in Eq. (5), we obtain the flitter equation :

[ Kgn + P Cgn + P2 Mgn – Fagn (Фп, P) ] qn = 0, (8)

which is a complex, nonlinear eigenvalue system in which the aerodynamic coefficient matrix Fagn (Фп, p) depends on the complex modes Фп and the unknown complex eigenvalue p.

When structural nonlinearities such as friction or free-play are to be intro­duced in the reduced coupled system Eq. (5), we need to keep some physical coordinates among the generalized coordinates qn. These "nonlinear” coordi­nates un|NL can be for example the displacements of the reference sector nodes located at the junction between the blade and the disk, where friction dampers are introduced. For this aim, the Craig and Bampton Craig and Bampton, 1968 projection basis Qn = [Фп, Фсп ] is used instead of the eigenmodes :

un = фп n + ^cn un|NL = Qn qn • (9)

Фп are the first mn complex normal modes of So with the cyclic symmetry conditions and with un|NL fixed; Фсп are the complex static constraint modes of S0 with the cyclic symmetry conditions and a unit displacement on one coordinate of un|NL, while the remaining coordinates of un|NL are fixed; and n is the vector of the modal generalized coordinates. By projecting Eq. (1) on Qn, we obtain a complex reduced coupled system similar to Eq. (5) whose size is the number of eigenmodes mn plus the number of the nonlinear coordinates.