Optimal mistuning for enhanced aeroelastic stability of transonic fans

Abstract

An inverse design procedure was developed for the design of a mistuned rotor. The design requirements are that the stability margin of the eigenvalues of the aeroelastic system be greater than or equal to some minimum stability margin, and that the mass added to each blade be positive. The objective was to achieve these requirements with a minimal amount of mistuning. Hence, the problem was posed as a constrained optimization problem. The constrained minimization problem was solved by the technique of mathematical programming via augmented Lagrangians. The unconstrained minimization phase of this technique was solved by the variable metric method of Broyden, Fletcher, and Shanno. The bladed disk was modelled as being composed of a rigid disk mounted on a rigid shaft. Each of the blades were modelled with a single tosional degree of freedom. Adamcyzk and Goldstein's linearized aerodynamic model for the unsteady moment coefficients in a supersonic cascade was applied at the typical section. The resulting non-self-adjoint eigenvalue problem is of the form Aq = XBq. The eigenvalues and eigenvectors of this eigenvalue problem were found by a fourth-order Runge-Kutta line integration of the derivatives of the eigenvalues and eigenvectors. It was shown that mass mistuning does not introduce damping into the system, and that a necessary but not sufficient condition for stability is that the blade be self damped. The results of the optimization showed that an optimally mistuned rotor can achieve a given stability margin for a much lower level of mistuning than alternate mistuning. However, it was shown that optimal mistuning is sensitive to errors in mistuning. Small errors in the implementation of optimal mistuning can severely reduce the gains in stability achieved by optimal mistuning. Alternate mistuning, on the other hand, is relatively insensitive to errors in mistune.