Aeroelastic Model

The starting point of this analysis are the fundamental equations of motion in physical space and in the time domain for a discretized system of coupled blades, for example conforming to a FEM approach. For a linear elastic sys­tem, these can be formally written as:

Мж + D ж + K x + Fc = F. (1)

Here M, D and K are the mass, damping and stiffness matrices, respectively,

while _Fc introduces the aerodynamic coupling between the blades and is de­pendent on the deflection and velocity of the blades. The damping is assumed to be of viscous type, i. e. the damping forces are proportional to the velocity.

The right hand side, F, contains the external forces acting on the structure, while the vector ж holds the displacement degrees-of-freedom for the finite el­ements of the complete model. For an accurate representation of the vibratory behavior, the number of degrees-of-freedom (DOF) for a typical cascade will have to be on the order of more than 10,000 times the number of blades. In the present method, a number of assumptions are introduced to arrive at a formu­lation that is at the same time accurate enough to give a good representation of the true aeroelastic behavior of the structure while still being simple enough to allow rapid analysis of a large number of configurations with variations in the dominant parameters.

The first assumption is that we are dealing with harmonic oscillations of the structure, so that the motion of each blade can be expressed using a complex exponential approach. Furthermore, in order to reduce the number of DOF, a modal approach is employed. It uses a small number nmodes of in-vacuo mode – shapes Ф of the individual blades as generalized coordinates. Thus, each blade of the cascade retains only a few degrees-of-freedom, which are characterized by the amplitudes ab, m of the respective modeshapes. While the amplitudes can differ between the individual blades, the underlying modeshapes are as-

sumed to be identical for all blades of the cascade under consideration. The motion xb of an arbitrary blade b is then approximated by

nmodes

X (r, $,z, t)= Xb(r, <£,z)eat = ab, m Ф m ЄМ

m=1

A = и + if

Here, the complex exponent A consists of a real part и that represents the angular frequency of oscillation and an imaginary part 7 that characterizes the evolution of the amplitude with time, representative of the effective damping present in the system under consideration. In this respect, a positive value of у implies an exponentially decaying oscillation amplitude, hence positive damping, while a negative value of 7 consequently yields exponentially grow­ing amplitudes, corresponding to negative damping, as in the case of a flitter instability. Accordingly, Eq. 1 can be approximated using a reduced set of eigenmodes as

[-A2M + iAD + (K + C)]a = F. (4)

In this equation, only generalized quantities are used. The vector a contains the complex amplitudes for all modes retained for the complete set of blades, hence for an annular cascade consisting of nblades individual blades, each of which retains nmodes modeshapes, the number of degrees-of-freedom in this reduced set of equations is nblades ■ nmodes. The matrix D holds the modal damping values that, just like the generalized masses and stiffnesses, can be assigned to each blade and mode individually. While the matrices M and D are diagonal, the stiffness matrix K can contain off-diagonal elements that are used to model mechanical coupling between the individual blades, as it is present through the disk on which the blades are mounted. The vector F in Eq. 4 contains the generalized external forces on the individual blades and modes

that result from the projection of the physical forces F on the eigenmodes Ф by

F = Ф*К

Here, the Ф* denotes the hermitian (conjugate-transpose) of the eigenvector matrix Ф. Similarly, the aerodynamic coupling forces have been transformed into the modal domain, additionally assuming that a linear aerodynamic ap­proach is valid. Then the modal forces due to the aerodynamic interactions Fc can be written as the product of the complex aerodynamic infhence coefficient matrix C and the modal amplitudes a:

The matrix C, in contrast to the other matrices, is usually fully populated. It contains the aerodynamic inflience coefficients that couple all blades and all modes of the cascade. In the present method, these inflience are calculated using a 3D linearized Euler solver that is an extension of the method described in [5], for further details refer to [6]. Using this reduced model, blade-to-blade variations can be readily introduced into the system by changing the general­ized properties mass, stiffness or damping assigned to any of the individual blade modes. Additionally, it is possible to add individually varying or con­stant mechanical coupling stiffness between different blades.