There is a need for a nonlinear flutter analysis method capable of predicting limit cycle oscillation in aeroelastic systems. A review is conducted of analysis methods and experiments that have attempted to better understand and model limit cycle oscillation (LCO). The recently developed method of nonlinear normal modes (NNM) is investigated for LCO calculation. Nonlinear normal modes were used to analyze a spring-mass-damper system with nonlinear damping and stiffness to demonstrate the ability and limitations of the method to identify limit cycle oscillation. The nonlinear normal modes method was then applied to an aeroelastic model of a pitch-plunge airfoil with nonlinear pitch stiffness and quasi-steady aerodynamics. The asymptotic coefficient solution method successfully captured LCO at a low relative velocity. LCO was also successfully modeled for the same airfoil with an unsteady aerodynamics model with the use of a first order formulation of NNM. A linear beam model of the Goland wing with a nonlinear aerodynamic model was also studied. LCO was successfully modeled using various numbers of assumed modes for the beam. The concept of modal truncation was shown to extend to NNM. The modal coefficients were shown to identify the importance of each mode to the solution and give insight into the physical nature of the motion. The quasi-steady airfoil model was used to conduct a study on the effect of the nonlinear normal mode's master coordinate. The pitch degree of freedom, plunge degree of freedom, both linear structural mode shapes with apparent mass, and the linear flutter mode were all used as master coordinates. The master coordinates were found to have a significant influence on the accuracy of the solution and the linear flutter mode was identified as the preferred option. Galerkin and collocation coefficient solution methods were used to improve the results of the asymptotic solution method. The Galerkin method reduced the error of the solution if the correct region of integration was selected, but had very high computational cost. The collocation method improved the accuracy of the solution significantly. The computational time was low and a simple convergent iteration method was found. Thus, the collocation method was found to be the preferred method of solving for the modal coefficients. / Ph. D.

When designing aircraft wings shapes, it is important to ensure that the flight envelope does not overlap with regions of flutter or Limit Cycle Oscillation (LCO). A quick assessment of these dynamic aeroelastic for various design candidates is key to successful design. Flutter based design requires the sensitivity of flutter parameters to be known with the respect of design parameters. Traditionally, frequency domain based methods have been used to predict flutter characteristics and its sensitivity. However, this approach is only applicable for linear or linearized models and cannot be applied to systems undergoing LCO or other nonlinear effects. Though the time accurate approach can be implemented to overcome this problem, it is computationally expensive. Also, the unsteady adjoint formulation for sensitivity analysis, requires the state and adjoint variables to be stored at every time step, which prohibitively increases the memory requirement. In this work, these problems have been overcome by implementing a time spectral method based approach to compute flutter onset, LCOs and their design sensitivities in a computationally efficient manner. The time spectral based formulation approximates the solution as a discrete Fourier series and directly solves for the periodic steady state, leading to a steady formulation. This can lead to the time spectral approach to be faster than the time accurate approach. More importantly, the steady formulation of the time spectral method also eliminates the memory issues faced by the unsteady adjoint formulation. The time spectral based flutter/LCO prediction method was used to predict flutter and LCO characteristics of the AGARD 445.6 wing and pitch/plunge airfoil section with NACA 64A010 airfoil. Furthermore, the adjoint based sensitivity analysis was used to carry out aerodynamic shape optimization, with an objective of maximizing the flutter velocity with and without constraints on the drag coefficient. The resulting designs show significant increase in the flutter velocity and the corresponding LCO velocity profile. The resulting airfoils display a greater sensitivity to the transonic shock which in turn leads to greater aerodynamic damping and hence leading to an increase in flutter velocity. / Doctor of Philosophy / When designing aircrafts, dynamic aeroelastic effects such as flutter onset and Limit Cycle Oscillations need to considered. At low enough flight speeds, any vibrations arising in the aircraft structure are damped out by the airflow. However, beyond a certain flight speed, instead of damping out the vibrations, the airflow accentuates these vibrations. This is known as flutter and it can lead to catastrophic structural failure. Hence, during the aircraft design phase, it must be ensured that the aircraft would not experience flutter during the flight conditions. One of the contribution of this work has been to come up with a fast and accurate method to predict flutter using computational modelling. Depending on the scenario, it is also possible that during flutter, the vibrations in the structure increase to a certain amplitude before leveling off due to interaction of non-linear physics. This condition is known as limit cycle oscillation. While they can arise due to different kinds of non-linearities, in this work the focus has been on aerodynamic non-linearities arising from shocks in transonic flight conditions. While limit cycle oscillations are undesirable as they can cause structural fatigue, they can also save the aircraft from imminent structural fracture and hence it is important to accurately predict them as well. The main advantage of the method developed in this work is that the same method can be used to predict both the flutter onset condition and limit cycle oscillations. This is a novel development as most of the traditional approaches in dynamic aeroelasticity cannot predict both the effects. The developed flutter/LCO prediction method has then been used in design with the goal of achieving superior flutter characteristics. In this study, the shape of the baseline airfoil is changed with the goal of increasing the flutter velocity. This enables the designed system to fly faster without addition of weight. Since the design has been carried out using gradient based optimization approach, an efficient way to compute the gradient needs to be used. Traditional approaches to compute the gradient, such as Finite Difference Method, have computational cost proportional to the number of design variables. This becomes a problem for shape design optimization, where a large number of design variables are required. This has been overcome by developing an adjoint based sensitivity analysis method. The main advantage of the adjoint based sensitivity analysis is that it its computational cost is independent of the number of design variables, and hence a large number of design variables can be accommodated. The developed flutter/LCO prediction and adjoint based sensitivity analysis framework was used to carry out shape design for a pitch/plunge airfoil section. The objective of the design process was to maximize the flutter onset velocity with and without constraints on drag. The resulting optimized airfoils showed significant increase in the flutter velocity.

This research study is the result of a 3 years CIFRE PhD thesis between the Airbus design office(Aircraft Control domain) and TéSA laboratory in Toulouse. The main goal is to propose, developand validate a software solution for the detection and diagnosis of a specific type of elevator andrudder vibration, called limit cycle oscillation (LCO), based on existing signals available in flightcontrol computers on board in-series aircraft. LCO is a generic mathematical term defining aninitial condition-independent periodic mode occurring in nonconservative nonlinear systems. Thisstudy focuses on the LCO phenomenon induced by mechanical freeplays in the control surface ofa civil aircraft. The LCO consequences are local structural load augmentation, flight handlingqualities deterioration, actuator operational life reduction, cockpit and cabin comfort deteriorationand maintenance cost augmentation. The state-of-the-art for freeplay induced LCO detection anddiagnosis is based on the pilot sensitivity to vibration and to periodic freeplay check on the controlsurfaces. This study is thought to propose a data-driven solution to help LCO and freeplaydiagnosis. The goal is to improve even more aircraft availability and reduce the maintenance costsby providing to the airlines a condition monitoring signal for LCO and freeplays. For this reason,two algorithmic solutions for vibration and freeplay diagnosis are investigated in this PhD thesis. Areal time detector for LCO diagnosis is first proposed based on the theory of the generalized likeli hood ratio test (GLRT). Some variants and simplifications are also proposed to be compliantwith the industrial constraints. In a second part of this work, a mechanical freeplay detector isintroduced based on the theory of Wiener model identification. Parametric (maximum likelihoodestimator) and non parametric (kernel regression) approaches are investigated, as well as somevariants to well-known nonparametric methods. In particular, the problem of hysteresis cycleestimation (as the output nonlinearity of a Wiener model) is tackled. Moreover, the constrainedand unconstrained problems are studied. A theoretical, numerical (simulator) and experimental(flight data and laboratory) analysis is carried out to investigate the performance of the proposeddetectors and to identify limitations and industrial feasibility. The obtained numerical andexperimental results confirm that the proposed GLR test (and its variants/simplifications) is a very appealing method for LCO diagnostic in terms of performance, robustness and computationalcost. On the other hand, the proposed freeplay diagnostic algorithm is able to detect relativelylarge freeplay levels, but it does not provide consistent results for relatively small freeplay levels. Moreover, specific input types are needed to guarantee repetitive and consistent results. Further studies should be carried out in order to compare the GLRT results with a Bayesian approach and to investigate more deeply the possibilities and limitations of the proposed parametric method for Wiener model identification.

Limit Cycle Oscillation is a type of aircraft wing structural vibration caused by the non-linearity of the system. The objective of this thesis is to provide a numerical study of this aeroelastic behavior. A CFD solver is used to simulate airfoils displaying such an aeroelastic behavior under certain airflow conditions. Two types of airfoils are used for this numerical study, including the NACA64a010 airfoil, and the supercritical NLR 7301 airfoil. The CFD simulation of limit cycle oscillation (LCO) can be obtained by using published flow and structural parameters. Final results from the CFD solver capture LCO, as well as flutter, behaviors for both wings. These CFD results can be obtained by using two different solution schemes, including the Roe and Zha scheme. The pressure coefficient and skin friction coefficient distributions are computed using the CFD results for LCO and flutter simulations of these two airfoils, and they provide a physical understanding of these aeroelastic behaviors.

The author examines nonlinear aeroelastic responses of air vehicle systems. Herein, the governing equations for a cantilevered configuration are developed and the methods of analysis are explored. Based on the developed nonlinear bending-bending-torsion equations, internal resonance, which is possible in future air vehicles, and the possible cause of limit cycle oscillations of aircraft wings with stores are investigated. The nonlinear equations have three types of nonlinearities caused by wing flexibility, store geometry and aerodynamic stall, and retain up to third-order nonlinear terms. The internal resonance conditions are examined by the Method of Multiple Scales and demonstrated by time simulations. The effect of velocity change for various physical parameters and stiffness ratio is investigated through bifurcation diagrams derived from Poinar´e maps. The dominant factor causing limit cycle oscillations is the stiffness ratio between in-plane and out-of-plane motion.

This work presents a method for building a three-dimensional mesh from two- dimensional topologically identical layers, for use in aeroelastic simulations. The method allows modeling of large deformations of the wing in both the span direction and deformations in the cord of the wing. In addition, the method allows for the modeling of wings attached to fuselages. The mesh created is a hybrid mesh, which allows cell clustering in the viscous region. The generated mesh is high quality and allows capturing of nonlinear uid structure interactions in the form of limit cycle oscillation.

The author develops a nonlinear longitudinal model of an aircraft modeled by rigid fuselage, tail, and wing, where the wing is attached to the fuselage with a torsional spring. The main focus of this research is to retain the full nonlinearities associated with the system and to perform gust load alleviation for the model by comparing the impact of a proportional-integral- lter nonzero setpoint linear controller with control rate weighting and a nonlinear Lyapunov-based controller. The four degree of freedom longitudinal system under consideration includes the traditional longitudinal three degree of freedom aircraft model and one additional degree of freedom due to the torsion from the wing attachment. Computational simulations are performed to show the aeroelastic response of the aircraft due to a gust load disturbance with and without control. Results presented in this thesis show that the linear model fails to capture the true nonlinear response of the system and the linear controller based on the linear model does not stabilize the nonlinear system. The results from the Lyapunov-based control demonstrate the ability to stabilize the nonlinear response, including the presence of an LCO, and emphasize the importance of examining the fully nonlinear system with a nonlinear controller.

The author examines nonlinear aeroelastic responses of air vehicle systems. Herein, the governing equations for a cantilevered configuration are developed and the methods of analysis are explored. Based on the developed nonlinear bending-bending-torsion equations, internal resonance, which is possible in future air vehicles, and the possible cause of limit cycle oscillations of aircraft wings with stores are investigated. The nonlinear equations have three types of nonlinearities caused by wing flexibility, store geometry and aerodynamic stall, and retain up to third-order nonlinear terms. The internal resonance conditions are examined by the Method of Multiple Scales and demonstrated by time simulations. The effect of velocity change for various physical parameters and stiffness ratio is investigated through bifurcation diagrams derived from Poinar´e maps. The dominant factor causing limit cycle oscillations is the stiffness ratio between in-plane and out-of-plane motion.

abstract: Recent studies of the occurrence of post-flutter limit cycle oscillations (LCO) of the F-16 have provided good support to the long-standing hypothesis that this phenomenon involves a nonlinear structural damping. A potential mechanism for the appearance of nonlinearity in the damping are the nonlinear geometric effects that arise when the deformations become large enough to exceed the linear regime. In this light, the focus of this investigation is first on extending nonlinear reduced order modeling (ROM) methods to include viscoelasticity which is introduced here through a linear Kelvin-Voigt model in the undeformed configuration. Proceeding with a Galerkin approach, the ROM governing equations of motion are obtained and are found to be of a generalized van der Pol-Duffing form with parameters depending on the structure and the chosen basis functions. An identification approach of the nonlinear damping parameters is next proposed which is applicable to structures modeled within commercial finite element software. The effects of this nonlinear damping mechanism on the post-flutter response is next analyzed on the Goland wing through time-marching of the aeroelastic equations comprising a rational fraction approximation of the linear aerodynamic forces. It is indeed found that the nonlinearity in the damping can stabilize the unstable aerodynamics and lead to finite amplitude limit cycle oscillations even when the stiffness related nonlinear geometric effects are neglected. The incorporation of these latter effects in the model is found to further decrease the amplitude of LCO even though the dominant bending motions do not seem to stiffen as the level of displacements is increased in static analyses. / Dissertation/Thesis / Masters Thesis Mechanical Engineering 2015

A conversão de vibração em energia elétrica tem sido investigada por diversos grupos de pesquisa na última década. A principal motivação é a prospecção de fontes alternativas de energia elétrica para sistemas eletroeletrônicos remotamente operados e com fontes limitadas de energia. Diferentes mecanismos de transdução são investigados na literatura para a coleta de energia, entretanto, o piezelétrico tem se destacado devido à densidade de energia que proporciona e também facilidade de uso. Uma alternativa promissora que começa a ser estudada por alguns grupos de pesquisas é a conversão de energia de oscilações aeroelásticas em energia elétrica. Apesar da natureza destrutiva da maioria dos fenômenos aeroelásticos, eles apresentam um grande potencial para o estudo de novos mecanismos e sistemas para coleta de energia. A conversão piezelétrica de energia a partir de oscilações aeroelásticas lineares tem sido investigada. Entretanto, a geração piezoaeroelástica de energia pode se tornar mais atrativa e prática se realizada a partir sistemas aeroelásticos não lineares. A conversão se daria a partir de oscilações persistentes e com amplitude limitada (oscilações em ciclo limite – LCO) ocorrendo em um amplo intervalo de velocidades de escoamento. Define-se o objetivo deste projeto como a investigação numérica da conversão piezelétrica de energia a partir de oscilações aeroelásticas não lineares. Um modelo por elementos finitos para placa plana com piezocerâmicas é desenvolvido, respeitando-se as hipóteses de uma placa de von Kàrmàn. O carregamento aerodinâmico não estacionário é determinado a partir do método de malha de dipolos e uma aproximação do domínio do tempo obtida a partir da formulação apresentada por Roger. Os resultados eletroaeroelásticos são apresentados para asas com diferentes razões de aspecto investigadas em uma ampla faixa de velocidades e considerando-se diversos valores de resistores no domínio elétrico. / The converting of vibration into usable electrical energy has been investigated by several researches groups in the last decade. The main motivation is the possibility of obtaining alternatives electrical energy sources to power electronic system remotely operated and with limited energy sources. Different transduction mechanism has been presented in the energy harvesting literature. However the piezoelectric has been gained more attention because not only of its power density but also its ease of use. A promissory alternative that is becoming studied is the converting of aeroelastic oscillation into electrical energy. Despite of the destructive nature of unstable aeroelastic phenomena (such as, flutter), they present a great potential to the study of innovative mechanism to harvest energy. Although the piezoelectric energy conversion using linear aeroelastic has been investigated in the literature, the use of non linear aeroelastic system can be more practical and attractive. The non linear aeorelastic harvesting occurs by persistent oscillation and with limited amplitudes (Limited Cycle Oscillation – LCO) and can be performed by considerable velocity interval greater than the linear flutter speed. The objective of this work is to investigate the energy harvesting by non linear aeroelastic oscillation. A finite element model of a thin plate (with piezoceramics) is developed), using the non linear hypothesis of von Karman. The unstable aerodynamic loading is obtained by a doublet-lattice method (DLM) and with its time domain conversion using the Roger approximation. The eletroaeroelastic results are presented for several wings with different aspect ratios, and with different resistance values in the electrical domain. The eletroaeroelastic results of the generator wing are investigated for several airspeed greater than its linear flutter speed.