The Use of Gramian Matrices for Aeroelastic Stability Analysis

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1. Introduction

Between various physical phenomena involving fluid-structure interaction, flutter is probably the most representative topic studied in engineering applications such as aircrafts and bridges. The flutter phenomenon is an interaction between structural dynamics and aerodynamics that results in divergent and destructive oscillations of motion [1].

In 1935, Theodorsen [2] proposed a method of flutter analysis in a discrete system by including aerodynamic forces in frequency domain and formulating the analysis as a complex eigenvalue problem. Hassig [3] proposes the pk-method where the unsteady aerodynamic matrix is represented by a function of the complex eigenvalues. Using an iterative algorithm the value of a reduced frequency converges to the imaginary part of a system eigenvalue. Chen [4] also proposes a flutter method including a first-order damping term into the equation of motion known as the g-method. According to the author, this method generalizes the k -methods and pk-methods for reliable damping prediction and has proved to be efficient in obtaining unlimited number of aerodynamic lag roots.

These methodologies, which are well established in the research and engineering community, were developed decades ago and have been used in the development of almost all flying commercial and military aircraft.

In this context, this paper proposes an alternative approach for detecting flutter using observability Gramian matrices. The proposed methodology is developed in time domain using state-space representation of the aeroelastic system. The elements of a Gramian matrix are related to the energy of vibration modes and can be seen as an improved observability matrix, introduced by Kalman et al. [5].

Gramian matrices have been used in the field of control engineering. Their fundamental concepts were proposed by Moore after introducing the balanced reduction for state-space models...