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.

Aquesta tesi es situa en el marc de la teoria qualitativa dels sistemes diferencials en el pla. Cada capítol conté un aspecte diferent. A la introducció, es dóna un resum dels resultats més coneguts i s'hi introdueix la notació que es fa servir al llarg de la tesi. En particular, descrivim el problema de la integrabilitat i alguns resultats sobre la determinació de l'estabilitat d'un punt singular o d'una òrbita periòdica a fi de presentar els darrers capítols. El problema de la integrabilitat es defineix com el problema de trobar la integral primera d'un sistema d'equacions diferencials en el pla i determinar la classe funcional a la qual pertany. Els Capítols 2 i 3 tracten el problema de la integrabilitat.En el Capítol 2 donem un resultat que permet trobar una expressió explícita per a una integral primera d'un cert tipus de sistemes polinomials. Mitjançant un canvi racional de variables, fem correspondre a una equació diferencial lineal homogènia de segon ordre: A2(x) w'(x) + A1(x) w'(x) + A0(x) w(x) = 0, els coeficients de la qual són polinomials, a un sistema diferencial polinomial pla. Provem que aquest sistema té un invariant per a cada solució arbitrària no nul·la w(x) de l'edo de segon ordre, que, quan w(x) és un polinomi, dóna lloc a una corba algebraica invariant. A més, donem una expressió explícita per a una integral primera del sistema construïda a partir de dues solucions independents de l'edo de segon ordre. Aquesta integral primera no és, en general, una funció Liouvilliana. Finalment, verifiquem que tots els exemples coneguts de famílies de sistemes quadràtics amb una corba algebraica invariant de grau arbitràriament alt es poden descriure mitjançant aquesta construcció (mòdul transformacions birracionals).En el Capítol 3, les corbes algebraiques invariants d'un sistema diferencial polinomial pla juguen el paper fonamental. Si un sistema diferencial polinomial pla té una corba algebraica invariant irreductible, aleshores els valors del seu cofactor en cadascun dels punts singulars no degenerats estan determinats. De fet, aquest valor es una combinació lineal a coeficients naturals dels valors propis associats al punt singular no degenerat. Aquests coeficients naturals es poden determinar completament en alguns casos depenent de la natura del punt singular. Així mateix, els punts de l'infinit també es poden tenir en compte. Un cop considerem el sistema en el pla projectiu complex, el grau d'una corba algebraica invariant esdevé un paràmetre del seu cofactor. Si considerem un sistema de grau d, aleshores té d^2 + d + 1 punts singulars (comptats amb la seva multiplicitat) i el cofactor d'una corba algebraica invariant té grau pel cap alt d-1. Procedim de la manera següent: prenem un polinomi de grau d-1 amb els seus d(d+1)/2 coeficients arbitraris i suposem que és el cofactor d'una corba algebraica invariant irreductible de grau n. Aleshores, imposem totes les condicions que ens donen els punts singulars no degenerats. En el cas general, imposem d^2 + d +1 condicions i, així, podem determinar completament el cofactor i el grau de la corba, l'existència de la qual es pot determinar resolent un sistema d'equacions lineal, o trobem una condició d'incompatibilitat. D'aquesta manera, en general, podem determinar l'existència de totes les corbes algebraiques invariants d'un sistema.El Capítol 4 tracta sobre l'estabilitat d'una òrbita periòdica d'un sistema diferencial pla. Suposem que f(x,y)=0 és una corba invariant irreductible amb cofactor que conté l'òrbita periòdica. Provem que les integrals sobre l'òrbita periòdica de la divergència i del cofactor coincideixen. Així, podem decidir sobre l'estabilitat de l'òrbita periòdica mitjançant la integració del cofactor sobre aquesta. En el Capítol 5, donem una aplicació dels resultats descrits en els Capítols 3 i 4. Considerem els sistemes quadràtics amb un cicle límit algebraic coneguts fins al moment de la redacció d'aquesta tesi. Aquest cicles límit algebraics estan continguts en corbes algebraiques invariants de graus 2, 4, 5 i 6 i algunes d'aquestes famílies de sistemes quadràtics son birracionalment equivalents. Aplicant el mètode descrit en el Capítol 3, mostrem que la corba algebraica invariant que conté el cicle límit es l'única corba algebraica invariant del sistema. Aprofitem aquest resultat per provar que aquests sistemes no tenen integral primera Liouvilliana. I aplicant la formula donada en el Capítol 4, provem que aquests cicles límit algebraics son hiperbòlics. El Capítol 6 tracta sobre l'estudi i les propietats de la funció període associada a un punt singular amb part lineal de tipus centre-focus. Com que el punt singular és sempre monodròmic, donada una secció transversal al flux amb el punt singular com a extrem, podem definir l'aplicació de Poincaré i la funció període associades a la secció. Diem que el punt és isòcron si podem trobar una secció tal que la seva funció període associada és constant. Aquesta definició generalitza la definició usual donada per centres a punts singulars qualssevol amb part lineal de tipus centre-focus. Caracteritzem aquesta propietat mitjançant simetries de Lie i formes normals, tot generalitzant els procediments coneguts per centres. Així mateix, donem un exemple d'una família de sistemes depenent d'un paràmetre real, tals que el seu origen és un punt singular amb part lineal de tipus centre-focus i que mai no és un punt isòcron. / Esta tesis se sitúa en el marco de la teoría cualitativa de los sistemas diferenciales en el plano. Cada capítulo contiene un aspecto distinto. En la introducción, se da un resumen de los resultados conocidos y se presenta la notación usada durante el resto de la tesis. En particular, se describe el problema de la integrabilidad y algunos resultados referentes a la determinación de la estabilidad de un punto singular o una órbita periódica con el fin de introducir los últimos capítulos. Definimos el problema de la integrabilidad como el problema de encontrar una integral primera para un sistema diferencial plano y determinar la clase funcional a la cual ésta debe pertenecer. Los Capítulos 2 y 3 tratan sobre el problema de la integrabilidad. En el Capítulo 2, obtenemos un resultado que permite encontrar una expresión explícita para una integral primera para un cierto tipo de sistema polinomial. Mediante un cambio racional de variable, hacemos corresponder una ecuación diferencial lineal homogénea de segundo orden: A2 (x) w'(x) + A1(x) w'(x) + A0(x) w(x) = 0, cuyos coeficientes son polinomios, a un sistema diferencial polinomial en el plano. Probamos que dicho sistema tiene un invariante para cada solución arbitraria no nula w(x) de la edo de segundo orden, que, en caso que w(x) sea un polinomio, da lugar a una curva algebraica invariante. Además, damos una expresión explícita de una integral primera para el sistema construida a partir de dos soluciones independientes de la edo de segundo orden. Esta integral primera no es, en general, una función Liouvilliana. Finalmente, verificamos que todos los ejemplos conocidos de familias de sistemas cuadráticos con una curva algebraica invariante de grado arbitrariamente alto se pueden describir mediante esta construcción (módulo transformaciones birracionales).En el Capítulo 3, las curvas algebraicas invariantes de un sistema diferencial plano polinomial juegan un papel fundamental. Si una curva algebraica invariante e irreducible existe para un sistema polinomial plano, entonces los valores de su cofactor en cada punto singular no degenerado están determinados. De hecho, este valor es una combinación lineal a coeficientes naturales de los valores propios asociados al punto singular no degenerado. Estos coeficientes naturales se pueden determinar completamente según la naturaleza del punto singular. Además, también podemos considerar los puntos del infinito. Una vez que el sistema se considera en el plano proyectivo complejo, el grado de una curva algebraica invariante deviene un parámetro de su cofactor. Si consideramos un sistema de grado d, entonces tiene d^2 + d + 1 puntos singulares (contados con su multiplicidad) y el cofactor de una curva algebraica invariante es un polinomio de grado a lo sumo d-1. Procedemos de la manera siguiente: tomamos un polinomio de grado d-1 con sus d(d+1)/2 coeficientes arbitrarios y suponemos que es el cofactor de una curva algebraica invariante e irreducible de grado n. Entonces, imponemos todas las condiciones dadas por los puntos singulares no degenerados. En el caso general, imponemos d^2 + d + 1 condiciones y, en consecuencia, determinamos completamente el cofactor y el grado de la curva, cuya existencia puede ser determinada resolviendo un sistema lineal de ecuaciones, o mostramos una condición de incompatibilidad. Por tanto, podemos determinar la existencia de todas las curvas algebraicas invariantes para un sistema general. El Capítulo 4 trata sobre la estabilidad de una órbita periódica de un sistema diferencial plano. Suponemos que f(x,y)=0 es una curva invariante e irreducible con cofactor que contiene la órbita periódica. Probamos que las integrales sobre la órbita periódica de la divergencia y del cofactor coinciden. De aquí que podamos deducir la estabilidad de una órbita periódica mediante la integración del cofactor sobre ésta. En el Capítulo 5, describimos una aplicación de los resultados dados en los Capítulos 3 y 4. Consideramos los sistemas cuadráticos con un ciclo límite algebraico conocidos hasta la redacción de esta tesis. Estos ciclos límite algebraicos están contenidos en curvas algebraicas invariantes de grados 2, 4, 5 y 6 y algunas de estas familias de sistemas cuadráticos son birracionalmente equivalentes. Aplicando el método descrito en el Capítulo 3, mostramos que no existe ninguna curva algebraica invariante excepto la que contiene el ciclo límite. Aprovechamos este resultado para mostrar que estos sistemas no tienen integral primera Liouvilliana. Y, aplicando la formula dada en el Capítulo 4, probamos que estos ciclos límite algebraicos son hiperbólicos. El Capítulo 6 trata sobre el estudio de las propiedades de la función periodo asociada a un punto singular con parte lineal de tipo centro-foco. Dada una sección transversal al flujo con dicho punto singular por extremo, podemos definir la aplicación de Poincaré y la función periodo asociadas a esta sección puesto que este punto es siempre monodrómico. Decimos que este punto es isócrono si podemos encontrar una sección tal que la función periodo asociada a ella sea constante. Esta definición generaliza la definición usual dada para centros a cualquier punto singular con parte lineal de tipo centro-foco. Caracterizamos esta propiedad mediante simetrías de Lie y formas normales, generalizando los procedimientos conocidos para centros. Además, damos un ejemplo de una familia de sistemas que dependen de un parámetro real, tales que el origen es un punto singular con parte lineal de tipo centro-foco y que nunca es un punto isócrono. / This thesis is situated in the framework of the qualitative theory of differential systems in the plane. Each chapter contains a different topic. In the introduction, a summary of known results is given and the notation used through the rest of the memory is presented. In particular, we describe the integrability problem and some results concerning the determination of the stability of a singular point or a periodic orbit in order to introduce the latest chapters. We define the integrability problem as the problem of finding a first integral for a planar differential system and determining the functional class it must belong to. Chapters 2 and 3 are concerned with the integrability problem. In Chapter 2, we obtain a result which allows to find an explicit expression for a first integral of a certain type of polynomial system. By means of a rational change of variable, we make correspond the homogenous second order linear differential equation: A2 (x) w'(x) + A1(x) w'(x) + A0(x) w(x) = 0, whose coefficients are polynomials, to a planar polynomial differential system. We prove that this system has an invariant for each arbitrary nonnull solution w(x) of the second-order ode, which, in case w(x) is a polynomial, gives rise to an invariant algebraic curve. In addition, we give an explicit expression of a first integral for the system constructed from two independent solutions of the second order ode. This first integral is not, in general, a Liouvillian function. Finally, we verify that all the known examples of families of quadratic systems with an invariant algebraic curve of arbitrarily high degree can be described by this construction (modulus birrational transformations). In Chapter 3, invariant algebraic curves of a planar polynomial differential system play the fundamental role. If an irreducible invariant algebraic curve for a planar polynomial differential system exists, then the values of its cofactor at each non-degenerate singular point are determined. In fact, this value is a linear combination with natural coefficients of the eigenvalues associated to the non-degenerate singular point. These natural coefficients can be completely determined in some cases depending on the nature of the singular point. Moreover, the points at infinity can also be taken into account. Once the system is considered in the projective complex plane, the degree of an invariant algebraic curve becomes a parameter of its cofactor. If we consider a system of degree d, then it has d^2 + d + 1 singular points (counted with multiplicity) and the cofactor of an invariant algebraic curve is a polynomial of degree at most d-1. We proceed as follows: we take a polynomial of degree d-1 with its d(d+1)/2 arbitrary coefficients and we assume that it is the cofactor of an irreducible invariant algebraic curve of degree n. Then, we impose all the conditions given by the non-degenerate singular points. In the general case, we impose d^2 + d + 1 conditions and, hence, we completely determine the cofactor and the degree of the curve, whose existence can be determined by solving a linear system of equations, or we show an incompatibility condition. Therefore, we can determine the existence of all the invariant algebraic curves of a general system.Chapter 4 is about the stability of a periodic orbit of a planar differential system. We assume that f(x,y)=0 is a real irreducible invariant curve with cofactor which contains the periodic orbit. We prove that the integrals over the periodic orbit of the divergence and the cofactor coincide. Hence, we can decide the stability of a periodic orbit by means of the integration of the cofactor over it. In Chapter 5, we describe an application of the results given in Chapters 3 and 4. We consider the quadratic systems with an algebraic limit cycle known until the composition of this thesis. These algebraic limit cycles are contained in invariant algebraic curves of degrees 2, 4, 5 and 6 and there are some of these families of quadratic systems which are birrationally equivalent one to the other. Applying the method given in Chapter 3, we show that there is no other irreducible invariant algebraic curve that the one which contains the limit cycle. We take profit from this result to show that these systems have no Liouvillian first integral. And applying the formula given in Chapter 4, we prove that these algebraic limit cycles are hyperbolic.Chapter 6 is devoted to the study of the properties of the period function associated to a singular point with linear part of centre-focus type. Given a section through the flow with such a singular point as endpoint, we can define the Poincaré map and the period function associated to this section since this point is always monodromic. We say that this point is isochronous if we can find a section such that the period function associated to it is constant. This definition generalizes the usual definition given for centres to any singular point with linear part of centre-focus type. We characterize this property by means of Lie symmetries and normal forms, generalizing the known procedures for centres. Moreover, we provide an example of a family of systems depending on a real parameter, such that the origin is a singular point with linear part of centre-focus type and which is never an isochronous point. / Cette thèse de doctorat traite sur la théorie qualitative des systèmes différentiels planaires. Chaque chapitre contient un sujet différent. Dans l'introduction, un sommaire des résultats connus est donné et la notation utilisée dans le reste du mémoire est présentée. En particulier, nous décrivons le problème de l'intégrabilité et quelques résultats concernant la détermination de la stabilité d'un point singulier ou d'une orbite périodique afin de présenter les derniers chapitres. Nous définissons le problème de l'intégrabilité comme le problème de trouver une intégrale première pour un système différentiel planaire et de déterminer la classe fonctionnelle à la quelle elle doit appartenir. Les Chapitres 2 et 3 traitent du problème de l'intégrabilité. Au Chapitre 2, nous obtenons un résultat permettant de trouver une expression explicite pour une intégrale première d'un certain type de système polynomial. Au moyen d'un changement rationnel de variables, nous faisons correspondre l'équation linéaire du deuxième degré: A2(x) w''(x) + A1(x) w'(x) + A0(x) w(x) = 0, dont les coefficients sont des polynômes, à un système différentiel polynomial planaire. Nous montrons que ce système a un invariant pour chaque solution arbitraire w(x) différent de zéro de l'équation considérée, qui, dans le cas où le w(x) serait un polynôme, est une courbe algébrique invariante. De plus, nous donnons une expression explicite d'une intégrale première pour le système construite à partir de deux solutions indépendantes de l'edo du deuxième degré. Cette intégrale première n'est pas, en général, une fonction de Liouville. En conclusion, nous vérifions que tous les exemples connus des familles des systèmes quadratiques avec une courbe algébrique invariante de degré arbitrairement élevé peuvent être décrits par cette construction (modulo des transformations birationnelles). Au Chapitre 3, les courbes algébriques invariantes d'un système différentiel polynomial planaire jouent le rôle fondamental. Si une courbe algébrique invariante et irréductible existe pour un système différentiel polynomial planaire, alors les valeurs de son cofacteur à chaque point singulier non dégénéré sont déterminées. En fait, cette valeur est une combinaison linéaire avec des coefficients naturels des valeurs propres associées au point singulier non dégénéré. Ces coefficients naturels peuvent être complètement déterminés dans certains cas selon la nature du point singulier. De plus, les points à l'infini peuvent également être pris en considération. Une fois que le système est considéré dans le plan projectif complexe, le degré d'une courbe algébrique invariante devient un paramètre de son cofacteur. Si nous considérons un système de degré d, alors il y a d^2 + d + 1 points singuliers (comptés avec sa multiplicité) et le cofacteur d'une courbe algébrique invariante est un polynôme de degré au plus d-1. Nous opérons comme suit: nous prenons un polynôme de degré d-1 avec ses d(d+1)/2 coefficients arbitraires et nous supposons que c'est le cofacteur d'une courbe algébrique invariante et irréductible de degré n. Nous imposons alors toutes les conditions données par les points singuliers non dégénérés. Dans le cas général, nous imposons d^2 + d + 1 conditions et, par conséquent, nous déterminons complètement le cofacteur et le degré de la courbe, dont l'existence peut être déterminée en résolvant un système linéaire d'équations, ou bien nous prouvons l'incompatibilité d'une condition. Par conséquent, nous pouvons déterminer l'existence de toutes les courbes algébriques invariantes d'un système général. Le sujet du Chapitre 4 est la stabilité d'une orbite périodique d'un système différentiel planaire. Nous supposons que f(x,y)=0 est une courbe invariante irréductible réelle avec cofacteur qui contient l'orbite périodique. Nous montrons que les intégrales sur l'orbite périodique de la divergence et le cofacteur coïncident. Par conséquent, nous pouvons déterminer la stabilité d'une orbite périodique en intégrant le cofacteur sur celle-ci.Dans le Chapitre 5, nous décrivons une application des résultats donnés aux Chapitres 3 et 4. Nous considérons les systèmes quadratiques avec un cycle limite algébrique connus jusqu'alors. Ces cycles limites algébriques sont contenus dans des courbes algébriques invariantes de degrés 2, 4, 5 et 6 et il existe certaines de ces familles de systèmes quadratiques qui sont birationnellement équivalentes. Appliquant la méthode exposée au Chapitre 3, nous prouvons qu'il n'y a aucune autre courbe algébrique invariante et irréductible différente à celle qui contient le cycle limite. Ceci nous permet de prouver que ces systèmes n'ont aucune intégrale première de Liouville. En appliquant la formule donnée au Chapitre 4, nous montrons que ces cycles limites algébriques sont hyperboliques. Le Chapitre 6 est consacré à l'étude des propriétés de la fonction de période associée à un point singulier dont la partie linéaire est de type centre-foyer. Etant donnée une section du flux avec tel point singulier comme point final, nous pouvons définir l'application de Poincaré et la fonction de période associée à cette section puisque ce point est toujours monodromique. Nous disons que ce point est isochronique si nous pouvons trouver une section telle que la fonction de période associée à elle est constante. Cette définition généralise la définition habituelle donnée pour des centres à n'importe quel point singulier dont la partie linéaire est de type centre-foyer. Nous caractérisons cette propriété au moyen des symétries de Lie et des formes normales, généralisant les procédures connues pour des centres. De plus, nous donnons un exemple d'une famille de systèmes avec un paramètre réel, telle que l'origine est un point singulier dont la partie linéaire est de type centre-foyer et qui n'est jamais un point isochronique.

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.

We consider a general class of finite dimensional deterministic dynamical systems with finitely many local attractors each of which supports a unique ergodic probability measure, which includes in particular the class of Morse–Smale systems in any finite dimension. The dynamical system is perturbed by a multiplicative non-Gaussian heavytailed Lévy type noise of small intensity ε > 0. Specifically we consider perturbations leading to a Itô, Stratonovich and canonical (Marcus) stochastic differential equation. The respective asymptotic first exit time and location problem from each of the domains of attractions in case of inward pointing vector fields in the limit of ε-> 0 has been investigated by the authors. We extend these results to domains with characteristic boundaries and show that the perturbed system exhibits a metastable behavior in the sense that there exits a unique ε-dependent time scale on which the random system converges to a continuous time Markov chain switching between the invariant measures. As examples we consider α-stable perturbations of the Duffing equation and a chemical system exhibiting a birhythmic behavior.

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

Abstract Digital delta-sigma modulators are used in a broad range of modern electronic sub-systems, including oversampled digital-to-analogue converters, class-D amplifiers and fractional-N frequency synthesizers. This work addresses a well known problem of unwanted spurious tones in the modulator’s output spectrum. When a delta-sigma modulator works with a constant input, the output signal can be periodic, where short periods lead to strong deterministic tones. In this work we propose means for guaranteeing that the output period will never be shorter than a prescribed minimum value for all constant inputs. This allows a relationship to be formulated between the modulator’s bus width and the spurious-free range, thereby making it possible to trade output spectrum quality for hardware consumption. The second problem addressed in this thesis is related to the finite accuracy of frequencies generated in delta-sigma fractional-N frequency synthesis. The synthesized frequencies are usually approximated with an accuracy that is dependent on the modulator’s bus width. We propose a solution which allows frequencies to be generated exactly and removes the problem of a constant phase drift. This solution, which is applicable to a broad range of digital delta-sigma modulator architectures, replaces the traditionally used truncation quantizer with a variable modulus quantizer. The modulus, provided by a separate input, defines the denominator of the rational output mean. The thesis concludes with a practical example of a delta-sigma modulator used in a fractional-N frequency synthesizer designed to meet the strict accuracy requirements of a GSM base station transceiver. Here we optimize and compare a traditional modulator and a variable modulus design in order to minimize hardware consumption. The example illustrates the use made of the relationship between the spurious-free range and the modulator’s bus width, and the practical use of the variable modulus functionality.