A generalized eigenvalue solution to the flutter stability problem with true damping: The p-L method

Abstract

An alternative solution to the flutter equation with a true damping representation is presented. The p-L method transforms the nonlinear eigenvalue problem into a linear generalized eigenvalue formulation by interpolating the nonlinear aerodynamic term with help of the Loewner and shifted-Loewner matrices, avoiding any kind of approximation. Subsequently an equivalent generalized state-space formulation in the time-domain is obtained, which stability can be determined by solving a standard generalized eigenvalue problem. The proposed method is shown to describe the aerodynamic term throughout the complex plane by analytic continuation of the interpolating rational functions over the imaginary axis, whereas expansions based on polynomial basis functions have a limited validity for excursions outside it. Further, the p-L method matches the generalized aeroelastic analysis method (GAAM) framework representing true damping but requiring only samples of the aerodynamic term along the imaginary axis, avoiding additional computations for non-zero damping. Unlike methods which solve the nonlinear eigenvalue problem whereby some roots of the flutter equation may be missed, the p-L method is able to find all roots at once. The method is applied to well-known flutter benchmark cases from the literature, namely, several two-dimensional flutter cases and the AGARD 445.6 wing aeroelastic benchmark case.