# Continuation Methods for Nonlinear Flutter

## Abstract

**:**

## 1. Introduction

## 2. Continuation Method

## 3. Flutter Equations

#### 3.1. Assumed Motion

**Φ**is typically a matrix of vibration modes, t is time, $s=\sigma +i\omega $ is the Laplace variable, ω is the frequency and oscillations are growing, neutral or decaying if the growth factor σ is positive, zero or negative, respectively.

#### 3.2. Equations

#### 3.3. Conversion to Real

#### 3.4. Matrix Parameterizations

#### 3.5. Normalization

#### 3.6. Continuation Formulations

#### 3.6.1. Constant η

#### 3.6.2. Variable η

## 4. Demonstration Problem

## 5. Searching for Limit Cycle Oscillations

#### 5.1. Point 0: Free-Vibration

#### 5.2. Process 1: V-σ-ω with ${\eta}_{0}=0$

#### 5.3. Process 2: σ-ω-η at $V=0$

#### 5.4. Processes 3 and 4: V-σ-ω

#### 5.5. Processes 5 and 6: σ-ω-η

#### 5.6. Stability of Equilibria

#### 5.7. Limit Cycles: V-ω-η Analysis at $\sigma =0$

#### 5.8. Optimal Path: V-σ-ω-η Analysis

## 6. Decreasing LCO Amplitudes

## 7. Bifurcation

## 8. Programming Considerations

## 9. Conclusions

## Conflicts of Interest

## Nomenclature

a | sonic velocity |

$\widehat{\mathit{D}}$, $\mathit{D}$ | complex and equivalent real dynamic matrices (Equations (13) and (16)) |

d | structural damping coefficient |

$\mathit{f}$ | vector of n real, nonlinear equations |

$\mathit{J}$ | Jacobian matrix of partial derivatives (Equation (5)) |

${\mathit{J}}_{i:}$, ${\mathit{J}}_{:\mathit{j}}$ | row i and column j of matrix $\mathit{J}$ |

$\mathit{M}$, $\mathit{K}$, $\mathit{G}$, | mass, stiffness, gyroscopic, viscous damping, |

$\mathit{V}$, $\mathit{A}$, $\mathit{T}$ | unsteady aero and control-system matrices (Equation (13)) |

M | $V/a$, Mach number |

${n}_{s}$ | the number of generalized coordinates |

p | $s/V$, complex reduced frequency |

${\parallel \mathit{x}\parallel}_{2}$ | 2-norm of vector $\mathit{x}=\sqrt{{\mathit{x}}^{T}\mathit{x}}$ |

q | $\rho {V}^{2}/2$, dynamic pressure |

$\widehat{\mathit{q}}$, $\mathit{q}$ | complex and equivalent real generalized coordinates (g.c.) (Equations (12) and (16)) |

$\mathit{Q}$, $\mathit{R}$ | QR factors of the transposed Jacobian (Equation (7)) |

${\mathbb{R}}^{n}$, ${\mathbb{R}}^{m\times n}$ | the real n-vectors and m by n matrices |

$\Re \left(\right),\Im \left(\right)$ | real and imaginary parts of a complex variable |

r, γ, δ | bilinear stiffness describing-function variables (Equation (19)) |

s | Laplace variable $\sigma +i\omega $ |

$\mathit{t}$ | tangent vector |

V | velocity (true airspeed) |

${V}_{max}$ | maximum velocity of interest (true airspeed) |

v | normalized velocity $V/{V}_{max}$ |

$\mathit{w}$ | projection vector (Equation (11)) |

$\mathit{x}$ | independent variables |

${\mathit{x}}_{j}^{i}$ | $i\mathrm{th}$ iteration at the $j\mathrm{th}$ continuation step (Equation (2)) |

${x}_{i}$ | $i\mathrm{th}$ element of vector $\mathit{x}$ |

$\widehat{\mathit{y}},\mathit{y}$ | complex and equivalent real eigenvector |

$\mathit{z}\left(t\right)$ | motion of points of the structure (Equation (12)) |

η | 2-norm of the generalized coordinates (Equation (15)) |

ω | oscillation frequency (imaginary part of the Laplace variable) |

Φ | transformation from generalized to physical coordinates |

ρ | fluid density |

σ | growth factor (real part of the Laplace variable) |

τ | arc length along a curve |

${\left(\right)}^{T}$ | transpose |

## References

- Allgower, E.L.; Georg, K. Numerical Continuation Methods: An Introduction; Springer Series in Computational Mathematics; Springer: Berlin, Germany, 1990; Volume 13. [Google Scholar]
- Meyer, E.E. Unified Approach to Flutter Equations. AIAA J.
**2014**, 52, 627–633. [Google Scholar] [CrossRef] - Shen, S.F. An Approximate Analysis of Nonlinear Flutter Problems. J. Aerosp. Sci.
**1959**, 26, 25–32. [Google Scholar] [CrossRef] - Gordon, J.T.; Meyer, E.E.; Minogue, R.L. Nonlinear Stability Analysis of Control Surface Flutter with Freeplay Effects. J. Aircr.
**2008**, 45, 1904–1916. [Google Scholar] [CrossRef] - Lee, C.L. An Iterative Procedure for Nonlinear Flutter Analysis. AIAA J.
**1986**, 24, 833–840. [Google Scholar] [CrossRef] - Breitbach, E.J. Flutter Analysis of an Airplane with Multiple Structural Nonlinearities in the Control System; NASA Langley Research Center: Hampton, VA, USA, 1980.
- Golub, G.H.; Van Loan, C.F. Matrix Computations, 4th ed.; The Johns Hopkins University Press: Baltimore, MD, USA, 2013. [Google Scholar]
- Gelb, A.; Vander Velde, W.E. Multiple-Input Describing Functions and Nonlinear System Design; McGraw-Hill: New York, NY, USA, 1968. [Google Scholar]
- Hassig, H.J. An Approximate True Damping Solution of the Flutter Equation by Determinant Iteration. J. Aircr.
**1971**, 8, 885–889. [Google Scholar] [CrossRef] - Chen, P.C. Damping Perturbation Method for Flutter Solution: The g-Method. AIAA J.
**2000**, 38, 1519–1524. [Google Scholar] [CrossRef] - Van Zyl, L.H.; Maserumule, M.S. Divergence and the p-k Flutter Equation. J. Aircr.
**2001**, 38, 584–586. [Google Scholar] [CrossRef] - Edwards, J.W.; Wieseman, C.D. Flutter and Divergence Analysis Using the Generalized Aeroelastic Analysis Method. J. Aircr.
**2008**, 45, 906–915. [Google Scholar] [CrossRef] - Rodden, W.P. MSC/NASTRAN Handbook for Aeroelastic Analysis; The MacNeal-Schwendler Corporation: Santa Ana, CA, USA, 2009; pp. 357–372. [Google Scholar]
- Bisplinghoff, R.L.; Ashley, H.; Halfman, R.L. Aeroelasticity; Addison-Wesley: Reading, MA, USA, 1955. [Google Scholar]
- Ruhe, A. Algorithms for the Nonlinear Eigenvalue Problem. SIAM J. Numer. Anal.
**1973**, 10, 674–689. [Google Scholar] [CrossRef] - Anderson, E.; Bai, Z.; Bischof, C.; Blackford, S.; Dongarra, J.; Du Croz, J.; Greenbaum, A.; Hammarling, S.; McKenney, A.; Sorensen, D. LAPACK Users’ Guide, 3rd ed.; Siam: Philadelphia, PA, USA, 1999. [Google Scholar]
- Meyer, E.E. Continuation and Bifurcation in Linear Flutter Equations. AIAA J.
**2015**, 53, 3113–3116. [Google Scholar] [CrossRef] - Rheinboldt, W.C.; Burkardt, J.V. A Locally Parameterized Continuation Process. ACM Trans. Math. Softw.
**1983**, 9, 215–235. [Google Scholar] [CrossRef] - LAPACK—Linear Algebra PACKage. Available online: https://www.netlib.org/lapack (accessed on 18 June 2016).
- Hajek, J. Qrupdate. Available online: https://sourceforge.net/projects/qrupdate/ (accessed on 2 September 2016).
- Langou, J. Determinant in QR factorization. Available online: http://icl.cs.utk.edu/lapack-forum/viewtopic.php?f=2&t=1741 (accessed on 14 May 2011).
- Griewank, A.; Walther, A. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd ed.; Siam: Philadelphia, PA, USA, 2008. [Google Scholar]
- Community Portal for Automatic Differentiation. Available online: http://www.autodiff.org (accessed on 2 September 2016).

**Figure 7.**Processes 2, 5 and 6: σ-ω-η at 3 normalized velocities (Green means stable, red means unstable).

**Figure 9.**Optimal path starting from V-σ-ω at $\eta =0$ (black curve with arrow indicating tracking direction) (Green means stable, red means unstable).

**Figure 10.**Optimal path: σ-ω, black curve with arrow indicating tracking direction (Green means stable).

**Figure 11.**LCO from the optimal path (black curve with arrow indicating tracking direction) (Green means stable, red means unstable).

**Figure 13.**Reduction of an LCO amplitude with control system parameters (Green means stable, red means unstable).

**Figure 14.**Bifurcation in a V-σ-ω process. Arrows indicate the direction of positive μ, the blue curve arises from the bifurcation (Green means stable, red means unstable).

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Meyer, E.E.
Continuation Methods for Nonlinear Flutter. *Aerospace* **2016**, *3*, 44.
https://doi.org/10.3390/aerospace3040044

**AMA Style**

Meyer EE.
Continuation Methods for Nonlinear Flutter. *Aerospace*. 2016; 3(4):44.
https://doi.org/10.3390/aerospace3040044

**Chicago/Turabian Style**

Meyer, Edward E.
2016. "Continuation Methods for Nonlinear Flutter" *Aerospace* 3, no. 4: 44.
https://doi.org/10.3390/aerospace3040044