Flutter analysis including structural uncertainties

Abstract

The common practice in industry is to perform flutter analyses considering the generalized stiffness and mass matrices obtained from finite element method (FEM) and aerodynamic generalized force matrices obtained from a panel method, as the doublet lattice method. These analyses are often re-performed if significant differences are found in structural frequencies and damping ratios determined from ground vibration tests compared to FEM. This unavoidable rework can result in a lengthy and costly process of analysis during the aircraft development. In this context, this paper presents an approach to perform flutter analysis including uncertainties in natural frequencies and damping ratios. The main goal is to assure the nominal system’s stability considering these modal parameters varying in a limited range. The aeroelastic system is written as an affine parameter model and the robust stability is verified solving a Lyapunov function through linear matrix inequalities and convex optimization.

Appendices

Appendix: State space submatrices

This appendix presents the submatrices $\bar{\mathbf{A}}_j$ previously shown in Eq. (9).

$$\begin{aligned} \bar{\mathbf{A}}_3&= \left[ q\mathbf{Q}_{m3} \; \cdots \; q\mathbf{Q}_{m(2+n_{lag})} \right] \end{aligned}$$

(22)

$$\begin{aligned} \bar{\mathbf{A}}_{4}&= \left[ \mathbf{I} \cdots \mathbf{I} \cdots \mathbf{I} \right] ^T, \;\; (n_{lag}+1)m \times m \;\;\;\; | \;\;\;\; \mathbf{I}, \;\; m \times m \end{aligned}$$

(23)

$$\begin{aligned} \bar{\mathbf{A}}_{5}&= \left[ 0 \right] ^{(n_{lag}+1)m \times m } \end{aligned}$$

(24)

$$\begin{aligned} \bar{\mathbf{A}}_{6}&= \nonumber \\&V \left( - \frac{1}{b} \right) \left[ \begin{array}{cccc} \mathbf{0} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} \\ \beta _1 \mathbf{I} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} \\ \cdots &{} \cdots &{} \cdots &{} \cdots \\ \mathbf{0} &{} \mathbf{0} &{} \cdots &{} \beta _{n_{lag}} \mathbf{I} \end{array} \right] , \;\; (n_{lag}+1)m \times n_{lag}m \end{aligned}$$

(25)

Aerodynamic forces in time domain

The state space model was obtained considering the unsteady aerodynamic forces written in time domain through the Roger’s rational function approximation shown in Eq. (26). Additional details can be found in reference [26].

$$\begin{aligned} \mathbf{Q}_{m}(s)&= \nonumber \\&\left[ \sum _{j=0}^{2}\mathbf{Q}_{mj}s^{j}\left( \frac{b}{V}\right) ^j +\sum _{j=1}^{n_{lag}}\mathbf{Q}_{m(j+2)}\left( \frac{s}{s+\frac{b}{V}\beta _{j}}\right) \right] \mathbf{u}_{m}(s) \end{aligned}$$

(26)

for which the parameters $\beta _{j}$ were previously chosen to keep the aeroelastic frequency and damping ratios computed by the pk-method for the nominal system. The generalized displacement vector $\mathbf{u}_{m}$ is used to define the state space vector $\mathbf{x}_{m} = \left\{ \dot{\mathbf{u}}_{m} \;\; \mathbf{u}_m \;\; \mathbf{u}_{am(1)} \ldots \mathbf{u}_{am(n_{lag})} \right\} ^T$. The Laplace variable is $s$ and $\mathbf{u}_{am(j)}$ is the $j$th vector of state of lags.