Abstract

The objective of this work is to investigate the dynamic behaviour of aero-elastic vibrations in the presence of non-linear stiffness such as free-play mechanism, softening or hardening stiffness. A closed loop dynamic system is proposed to represent the phenomenon of flow-structure interaction. In this approach a transfer function for generating the aerodynamic forces based on the structural response is constructed in the feedback loop of the dynamic system with the aid of Padé rational function. The effects of the non-linear factors therefore can be included conveniently in time domain simulation and the stability of limit cycle oscillations (LCO) can be analysed accurately.

Keywords: LCO, non-linear structures, flow structure interactions, aero-elasticity, self-excited vibration, binary classical flutter

## 1. Introduction

Aero-elasticity is a multi-physics discipline that involves the loads of aerodynamics, elastic and inertial generated by the motion of structure. One of the most important phenomena in this field is flutter regarding to its harmful effect to the structure. This flow-induced vibration under certain conditions can be self-excited and divergently unstable. In aerospace industry the boundary of flutter instability is usually determined by V-g method, a computational technique in frequency domain based on the balance of energy of the oscillating wing and the flow by maintaining a harmonic function of the aero-elastic response. In this method explained remarkably by Stanciu et al. [1], the critical velocity of the flow is determined by solving the complex eigen value problem of the aero-elastic system. Although the method can give accurate results it is only effective for linear cases. However in real aircraft there are non-linear factors of structure such as free-play, hysteretic and large deformation that need to be taken into account. Trickey et al. [2] observed that certain cases in regards to excessive LCO found in some Boeing and Airbus aircrafts and stated that the characterization and explanation of this non-linear vibration were important for fatigue and maintenance issues. In their research, the methods of non-linear dynamics were developed for these purposes and they proposed a novel system-identification technique to generate an approximation of LCO to be used for online monitoring of dynamic behaviour close to bifurcation condition [8].

Pereira et al. [3] showed an example of LCO due to the existence of hardening nonlinearity of wing stiffness in pitching of F-16 aircraft that caused persistent aero-elastic problems. Therefore the knowledge and comprehension of non-linear aero-elasticity are of increasing importance in aircraft design. In their work an investigation on the combined influence of hardening and free-play nonlinearities on the bifurcation response was carried out.

A document regarding the missing of MH370 Boeing 777 is also concerned about the phenomena of aero-elasticity. The failure analysis of the right-side flaperon that was found in French territory's Reunion Island [4] on 2015, reported that flutter (LCO) caused to repetitive loading which in turn imparted stress fatigue in the primary aluminum alloy attachment components.

This work has an intention to simulate numerically the interaction of flowstructure as a dynamic system by arranging the structure part as a principle plant and the aerodynamic part as a feedback loop subsystem. The analysis of structural response in time domain enables to insert the nonlinearities conveniently. The part of structures is reconstructed in a form of block-diagram representing a dynamic system where the inertial loads are expressed explicitly as a result of the elastic loads and frictions generated in the progressing structure response due to external aerodynamic excitations, while the part of aerodynamics is arranged as a feedback-loop transfer function activated by the structural response. For this purpose, the unsteady aerodynamic forces calculated by using singularity method in frequency domain have to be converted to Laplace variable s by using Padé's approximation rational function. Botez et al. [5] in conducting flutter analysis of CL-604 Bombardier, used a least-squares technique utilizing certain number of lagging-terms, and the approximation showed the best aerodynamic forces conversion from frequency into Laplace domain in terms of execution time and precision.

In analysing LCO on mechanical system in general where there is involvement of various physical parameters such as non-linear stiffness, hysteretic and free-play, and more specifically the influence of damping to the stability of the oscillations, Sinou and Jézéquel [6] proposed to employ a two-degree-of-freedom model for the sake of simplicity. With the same spirit, in this study we use a pitch-plunge two dimensional wing-section model in analysing the effects of structural nonlinearities on a binary classical flutter.

2.2 Unsteady aerodynamic model

Two-degree-of-freedom aero-elastic model.

steady of the lift coefficient CQS

CQS <sup>L</sup> <sup>¼</sup> <sup>∂</sup>Cl

tion or intrusion.

attack:

Figure 1.

presented as:

139

Theodorsen's unsteady aerodynamic model as explained by Brunton and Rowly [7] excellently is used in this work. This method analyses the motions of the aerofoil in frequency domain and it assumes that the amplitudes are small. In the analysis the aerofoil is considered thin, the flow is inviscid incompressible with no separa-

In this method the frequencies of the harmonic oscillating motions are considered relatively slow therefore the transversal and rotational velocities of the aerofoil contribute as an additional angle of attack to the total lift. As a result the quasi-

> h U<sup>∞</sup>

<sup>þ</sup> <sup>b</sup> <sup>1</sup>

In thin aerofoil theory the lift gradient can be considered equals to 2π, the vortex singularity is located at the aerodynamic center (a quarter of the chord from the leading-edge) and the downwash velocity is focused at three quarter of the chord. The aerodynamic loading consisted of lift and pitching moment can be

<sup>ρ</sup>U<sup>2</sup> clC kð Þ <sup>C</sup>QS

<sup>2</sup> lC kð Þ <sup>C</sup>QS

!

<sup>2</sup> � <sup>a</sup> � � α\_

<sup>∂</sup><sup>α</sup> <sup>α</sup> <sup>þ</sup> \_

Analysing Non-Linear Flutter Vibrations Using System Dynamic Approach

DOI: http://dx.doi.org/10.5772/intechopen.85426

<sup>L</sup> <sup>¼</sup> <sup>1</sup> 2

Mac <sup>¼</sup> <sup>1</sup> 2 ρU<sup>2</sup> c

<sup>L</sup> can be expressed proportional to the total angle of

U<sup>∞</sup>

<sup>L</sup> (3)

<sup>L</sup> (4)

(2)
