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) c i = ith off-diagonal quadratic stability coef cient, Eqs.

Design of Beneficial Wave Dynamics for Engine Life and Operability Enhancement

“…Recent work has revealed the importance of skew-symmetric coupling in wave-equation systems [7,8] and how mistuning can enhance the stability of such systems. This phenomena is also common in suppression of blade flutter instabilities [10,60,58,70]. Additionally, recent work has focussed on analysis of heterogeneous distributed systems [24], [40].…”

“…39,8]. Often, the dynamic models of these systems are characterized by a large number of similar subsystems through neighboring coupling, and the coupling of the subsystems may have aperiodic or random patterns [70,60]. As a result, stability analysis of such dynamical systems becomes a challenge.…”

Optimization of a Quadratic Function with a Circulant Matrix

“…Moreover, judiciously applied intentional mistuning has been shown to improve performance by increasing the region of stability (via extension of the flutter boundaries which delineate unstable blade vibration). Thus, enhanced robust performance is achieved within existing manufacturing tolerances [5].…”

Analyzing Mistuning of Bladed Disks by Symmetry and Reduced-Order Aerodynamic Modeling

“…'z/ where p`D exp [2¼ i`=r ]. This implies that if we know the rst blade dynamics X 1 .z/ for all mistuning z, then we know the response for all other blades by symmetry.…”

“…From a practical point of view, we are concerned primarily with small blade deformations and so any nonlinear model reduces to the standard linear problem P x D M.z/x C B`.z/e i`Ät (1) where x D .x 1 ; x 2 ; : : : ; x r / 2 R r m is the state vectorwith x i 2 R m corresponding to aerodynamic and structural states for the i th blade. Here r is the number of blades, m is the states per blade.…”