# Generalized Quantitative Stability Analysis of Time-Dependent Comprehensive Rotorcraft Systems

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## Abstract

**:**

## 1. Introduction

- alternating loads that excite the fuselage, and
- alternating system properties leading to a time-dependent system.

## 2. Method

#### 2.1. Lyapunov Characteristic Exponents

- First, the time response of the system converges to zero, which happens when all eigenvalues have a negative real part.
- Second, the system responds in a divergent manner when some eigenvalues have a positive real part.
- Third, a periodic orbit can be the resulting motion in the presence of purely imaginary eigenvalues.

#### 2.2. Application to LTP Problems

#### 2.3. Numerical Estimation of LCEs

#### 2.4. Computation of State Transition Matrix

#### 2.5. Analytical Sensitivity of LCEs

#### 2.5.1. Sensitivity of QR Decomposition

#### 2.5.2. Sensitivity of the State Transition Matrix

## 3. Numerical Results

#### 3.1. Verification with an Analytical Solution

#### 3.2. Complex LTP Rotorcraft Model

#### 3.2.1. Model Description

- all six rigid body modes (Fore/Aft, Lateral, Plunge, Roll, Pitch, and Yaw);
- flight mechanics derivatives of the fuselage determined by fuselage/wing-body, horizontal tail, and the vertical tail;
- ten elastic airframe modes captured at airframe and connection locations (such as airframe rotor connection); additionally, $1.5$% modal damping was also added;
- three bending modes of the rotor in multiblade coordinates, formulated using a linerazized finite volume approach [26];
- three main rotor servo-actuators modelled as transfer function from the force applied by controls (${f}_{c}$) and requested displacement (${x}_{c}$) to the servoactuator displacement (${x}_{s}$), namely ${x}_{s}={H}_{x}{x}_{c}+{H}_{f}{f}_{c}$;
- lead-lag dampers for the main rotor blades.

#### 3.2.2. Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

LCE | Lyapunov Characteristic Exponents |

LTI | Linear, Time-Invariant |

LTP | Linear, Time-Periodic |

LTV | Linear, Time-Variant |

STM | State Transition Matrix |

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**Figure 1.**Comparison of the characteristic exponents obtained by analytical solution and numerical method (Floquet).

**Figure 2.**Comparison of the sensitivity characteristic exponents obtained by analytical solution and numerical method (Floquet).

**Figure 3.**Characteristic exponents estimated by Lyapunov theory (LCEs) and Floquet multipliers, whole range. The lightly damped modes are marked with blue.

**Figure 4.**Characteristic Exponents estimated by Lyapunov theory (LCEs) and Floquet multipliers, zoom of lightly damped modes from Figure 3. The lightly damped blade lead-lag mode is marked in green.

**Figure 5.**Characteristic exponents estimated by Lyapunov theory (LCEs) and Floquet multipliers, corresponding to the lightly damped blade lag mode, isolated and zoomed from Figure 4.

**Figure 6.**Sensitivity estimates of characteristic exponents from Lyapunov theory (LCEs), Floquet Analysis, and finite differences, corresponding to the lightly damped blade lag mode of Figure 5.

**Table 1.**AS-330 PUMA general characteristics [25].

Parameter | Value | Units |
---|---|---|

Helicopter | ||

Gross Weight | 7400 | kg |

Max Speed | 140 | kn |

Main Rotor | ||

Number of blades | 4 | |

Radius | 7.49 | m |

Solidity | 0.0913 | (n.d.) |

Lock number | 8.70 | (n.d.) |

Speed | 270 | rpm |

Flap frequency | 1.03 | /rev |

Lag Frequency | 0.26 | /rev |

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**MDPI and ACS Style**

Tamer, A.; Masarati, P.
Generalized Quantitative Stability Analysis of Time-Dependent Comprehensive Rotorcraft Systems. *Aerospace* **2022**, *9*, 10.
https://doi.org/10.3390/aerospace9010010

**AMA Style**

Tamer A, Masarati P.
Generalized Quantitative Stability Analysis of Time-Dependent Comprehensive Rotorcraft Systems. *Aerospace*. 2022; 9(1):10.
https://doi.org/10.3390/aerospace9010010

**Chicago/Turabian Style**

Tamer, Aykut, and Pierangelo Masarati.
2022. "Generalized Quantitative Stability Analysis of Time-Dependent Comprehensive Rotorcraft Systems" *Aerospace* 9, no. 1: 10.
https://doi.org/10.3390/aerospace9010010