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Eigenvalue Analysis
In the absence of external forces, the prime interest is on the eigenmodes and eigenvalues of the cascade, describing the flitter stability and the free vibration behavior of the cascade. For this purpose, Eq. 4 can be reformulated as
[M-1(K + C)+ iM-1 DA – EA2]a = 0 (7)
Being a non-linear eigenvalue problem, this is fairly inconvenient to solve numerically. To overcome this, it is assumed that the aeroelastic eigenvalues A do not differ much from the in-vacuo eigenvalues A0. Then we can approximate Eq. 7 as
[M-1(K + C)+ iM-1DAo – EA2]a = 0. (8)
In this approach, no assumption concerning the distribution of damping in the cascade is made, so that also the effect of damping mistuning can be studied. On the other hand, the approach can only be expected to yield valid results as long as the computed eigenvalues are similar to the in-vacuo eigenvalues of the individual blades. For practical purposes, this condition is usually well met, as discussed in more detail in [6]. To evaluate the aeroelastic stability of a cascade in the presence of aerodynamic and structural coupling as well as structural mistuning, we thus have to solve a linear eigenvalue problem of nblades • nmodes DOF, which can be done very quickly and efficiently with standard numerical solvers even for a large number of mistuned configurations.
