Symmetry Approach to Extension of Flutter Boundaries via Mistuning

Abstract

A general framework is presented for analyzing and optimizing stability increases resulting from mistuning. The framework given is model independent and is based primarily on symmetry arguments. Dif � cult practical issues are transformed to tractable mathematical questions. It is shown that mistuning analysis reduces to a block circular matrix eigenvalue/vector problem that can be solved ef � ciently even for large problems. Similarly, the optimization becomes a standard linear constraint quadratic programming problem and can be solved numerically. Because the methods given are model-independent, they can be applied to various models and allow the researcher to easily conclude which models accurately capture mistuning and which do not. A simple quasisteady model for � utter in a cascade is used to illustrate and validate results in this paper. Nomenclature a = linear stability coef � cient, Eqs. (13), (14), and (35) b = diagonal quadratic stability coef � cient, Eqs. (13), (14), and (36) ci = ith off-diagonal quadratic stability coef � cient, Eqs. (13), (14), and (37) f = f (x, U, z), � utter model, Eq. (1) h = h(U, z), negative of minimum damping, Eq. (3) k = r/2 when r even, (r 1)/2 if r odd M = M(U, z), linearization matrix in Eq. (15) m = number of states per blade r = number of blades S = stability extension matrix, Eq. (14) s = s(z) stability extension, Eq. (5) U = loading parameter in Eq. (1) Ucrit = Ucrit(z), instability inception loading, Eq. (4) Ui, Vi = ith left and right eigenvectors of M [Ucrit(0), 0] in Eqs. (38) and (39) x = (x1, x2,..., xr) state vector, xi blade, Eq. (1) x0 = x0(U, z), relevant equilibrium point of model, Eq. (1) z = (z1, z2,..., zr) mistuning vector, zi mistuning