<p>High performance fighter aircraft such as the F-16 experience aeroelastic Limit Cycle Oscillations (LCO) when they carry certain combinations of under-wing stores. This `store-induced LCO' causes serious problems including airframe fatigue, pilot discomfort and loss of operational effectiveness. The usual response has been to restrict the stores carriage envelope based on flight test experience, and accept the accompanying reduction in mission performance.</p><p>Although several nonlinear mechanisms - structural as well as aerodynamic, have been proposed to explain the LCO phenomenon, their roles are not well understood. Consequently, existing models are unable to predict accurately AND reliably the most critical LCO properties, namely onset speed and response level. On the other hand, the more accurate Computational Fluid Dynamics (CFD) based time marching methodology yields results at much greater expense and time. Clearly, there is a critical need to establish methods that are more rapid while providing accurate predictions more in line with flight test results than at present. Such a capability will also aid in future aircraft design and usage.</p><p>This work was undertaken to develop a better understanding of nonlinear aeroelastic phenomena, and their relation to classical flutter and divergence, with a particular focus on store-induced LCO in high performance fighter aircraft. The following systems were studied: (1) a `simple' wing with a flexible and nonlinear root attachment, (2) a `generic' wing with a flexible and nonlinear wing-store attachment and (3) the F-16 aircraft, again with nonlinear wing-store attachments.</p><p>While structural nonlinearity was present in all cases, steady flow aerodynamic nonlinearity was also included in the F-16 case by the use of a Computational Fluid Dynamics model based on the Reynolds Averaged Navier Stokes (RANS) equations. However, dynamic linearization of the CFD model was done for the present computations. The computationally efficient Harmonic Balance (HB) nonlinear solution technique was a key component of this work, with time marching simulations and closed form solutions being used selectively to confirm the findings of the HB solutions. The simple wing and the generic wing were both modeled as linear beam-rods whose displacements were represented using the primitive modes method. The wing aerodynamic model was linear (quasi-steady for the simple wing and based on the Vortex Lattice Method for the generic wing), and the store aerodynamics were omitted.</p><p>The presence of a cubic restoring force (of hardening or softening type, in stiffness or in damping) at the root of the simple wing led to several interesting results and insights. Next, various nonlinear mechanisms including cubic restoring force, freeplay and friction were introduced at the wing-store attachment of the generic wing and these led to a still greater variety in behavior. General relationships were established between the type of nonlinearity and the nature of the resulting response, and they proved very useful for tailoring the F-16 study and interpreting its results.</p><p>The Air Force Seek Eagle Office/Air Force Research Laboratory provided a modal structural model of an LCO-prone store configuration of the F-16 aircraft with stores included. In order to investigate a range of stores attachment configurations, the analysis required modification of the stiffness and damping of the wing-store attachment. Since the Finite Element model of the wing and store structure was not available, the modification was achieved by subtracting the store and adding it back with the necessary changes to the store or attachment using a dynamic decoupling/coupling technique. The modified models were subjected to flutter/LCO analysis using the Duke Harmonic Balance CFD RANS solver, and the resulting flutter boundaries were used in combination with the HB method to derive LCO responses due to the wing-store attachment nonlinearity.</p><p>Comparisons were made between the simulation results and the F-16 flight test LCO data. While multiple sources of nonlinearity are probably responsible for the wide range of observed LCO behavior, it was concluded that cubic softening stiffness and positive cubic damping were the more likely structural mechanisms causing LCO, in addition to nonlinear aerodynamics.</p> / Dissertation

Complex nonlinear aspects of aeroelastic phenomena include unsteady nonlinear aerodynamic loads, structural nonlinearities, as well as nonlinear couplings between the flow and the structural response. Nonlinearities in aerodynamic loads originate from unsteady shocks and/or flow separation. Structural nonlinearities are geometric, or a result of free play. Nonlinear fluid structure couplings result from nonlinear resonance between the aerodynamic load and structural modes. Under different conditions, one or a combination of these aspects could yield flutter or Limit Cycle Oscillations (LCO). The overall goal of this work is to develop the capabilities to quantify the role that these different nonlinear mechanisms could play in observed flutter and LCO. The realization of such a goal would help in providing a benchmark for the detection of nonlinear aeroelastic instabilities and possibly effective means for obtaining improved performance and reduced uncertainties through operation beyond conventional boundaries that are based on linear analysis. Additionally, this effort will provide a benchmark for the validation of computational methodologies. In this thesis, wavelet-based higher order spectra are applied to identify different nonlinear aeroelastic phenomena as encountered in two experiments. First, the analysis is applied to a set of experiments involving a flexible semispan model (FSM) of a High Speed Civil Transport (HSCT) wing configuration conducted by Silva et al. (Experimental Steady and Unsteady Aerodynamic and Flutter Results for HSCT Semispan Models; AIAA/ASME/ASCE/AHS/ASC 41st Structures, Structural Dynamics, and Materials Conference, 2000). The interest is in the identification of nonlinear aeroelastic phenomena associated with a high dynamic response region which was measured over a large range of dynamic pressures around Mach number 0.98. At the top of this region is a ``hard'' flutter point that resulted in the loss of the model. The results show that ``hard'' flutter is related to intermittent nonlinear coupling between the shock motion and large amplitude structural motions. Second, the analysis is applied to identify nonlinear aspects of LCO encountered during test flights of an F-16 aircraft. The results show quadratic and cubic couplings in the acceleration signals of the under-wing launchers and high quadratic coupling levels between flaperon motions and wing oscillations. The implications of applying these techniques in the capacity of a ``flutterometer'' are also discussed. / Ph. D.

Presence of nonlinearities may lead to limit cycle oscillations (LCOs) in aeroelastic systems. LCOs can result in fatigue in wings leading to catastrophic failures. Existence of LCOs for velocities less than the linear flutter velocity has been observed during flight and wind tunnel tests, making such subcritical behavior highly undesirable. The objective of this dissertation is to investigate the existence of subcritical LCOs in aeroelastic systems and develop state feedback controllers to suppress them. The research results are demonstrated on a two degree of freedom airfoil section model with stiffness nonlinearity. Three different approaches are developed and discussed. The first approach uses a feedback linearization controller employing the aeroelastic modal coordinates. The use of modal coordinates results in a system which is linearly decoupled making it possible to avoid cancellation of any linear terms when compared to existing feedback linearization controllers which use the physical coordinates. The state and control costs of the developed controller are compared to the costs of the traditional feedback linearization controllers. Second approach involves the use of nonlinear normal modes (NNMs) as a tool to predict LCO amplitudes of the aeroelastic system. NNM dynamics along with harmonic balance method are used to generate analytical estimates of LCO amplitude and its sensitivities with respect to the introduced control parameters. A multiobjective optimization problem is solved to generate optimal control parameters which minimize the LCO amplitude and the control cost. The third approach uses a nonlinear state feedback control input obtained as the solution of a multiobjective optimization problem which minimizes the difference between the LCO commencement velocity and the linear flutter velocity. The estimates of LCO commencement velocity and its sensitivities are obtained using numerical continuation methods and harmonic balance methods. It is shown that the developed optimal controller eliminates any existing subcritical LCOs by converting them to supercritical LCOs. / Ph. D.

Thesis (PhD (Biochemistry))--University of Stellenbosch, 2009. / ENGLISH ABSTRACT: With the recent advances in the field of molecular biology, there is an increased need to integrate data on the various constituents of the cell in kinetic models that can predict and describe cellular behavior. When working towards a description of the entire cell using such kinetic models, the question arises: How do we compare different models for a given biological network? This is the central question addressed in my thesis and I developed and applied mathematical and computational methods for comparing dozens of existing models of erythrocyte and yeast glycolysis. To compare the steady-state behavior in models of erythrocyte glycolysis, I focussed on the function of the pathway, which is to supply the cell with Gibbs-free energy (γ- phosphate of ATP). I used supply-demand analysis in the framework of metabolic control analysis to make this comparison, which revealed that the ATP concentrations were homeostatically buffered at varying supply rates. I also applied this approach to compare steady-state behavior in models of yeast glycolysis, finding that they were not necessarily optimized for homeostatic maintenance of the ATP concentration and that in models for this organism the rate of ATP production is often determined by the supply reactions of glycolysis. In addition, I tested whether a kinetic model can describe novel behavior if it is adjusted to conditions different from those for which the model was originally constructed. More specifically, using a model of steady-state yeast glycolysis, I showed that small adjustments to the original enzyme concentrations are enough to obtain an oscillating model, which shows a remarkable resemblance to the experimentally observed oscillations. Importantly, some of these enzyme concentrations changes are known to occur during the pre-treatment of the cells which is necessary to obtain oscillatory behavior. To the best of my knowledge, the resulting model is the first detailed kinetic model that describes the experimentally observed strong synchronization of glycolytic oscillations in yeast populations. To analyze the dynamic behavior of yeast glycolytic models and to compare different models in terms of dynamics, I introduced a framework used in physics and engineering to create a vector based, two dimensional graphical representation of the oscillating metabolites and reactions of glycolysis. Not only was it possible to make a concise comparison of the set of models, but with the method I could also quantify the contribution of the interactions in the network to the transduction of the oscillations. Furthermore I could distinguish between different mechanisms of oscillation for each of the models, and demonstrated how the framework can be used to create such representations for experimental data sets. / AFRIKAANSE OPSOMMING: Met die onlangse vooruitgang in die veld van molekulere biologie, is daar ?n toenemende behoefte om data rakende die verskeie komponente van die sel in kinetiese modelle te integreer, om sodanig selgedrag te voorspel en te beskryf. As daar gepoog word om ’n beskrywing van die sel as geheel te verkry d.m.v. sulke kinetiese modelle, onstaan die vraag: Hoe vergelyk ons verskillende modelle van ’n gegewe biologiese netwerk? Dit is die sentrale vraag wat my tesis aanspreek en ek het wiskundige en numeriese metodes ontwikkel en toegepas om talle bestaande modelle van gis- en eritrosietglikolise te vergelyk. Om die bestendige-toestand gedrag in modelle van eritrosietglikolise te vergelyk, het ek gefokus op die funksie van die padweg, naamlik om die sel met Gibbs-vrye energie (γ-fosfaat van ATP) te voorsien. Ek het vraag-aanbod analiese in die raamwerk van metaboliese kontrole analiese gebruik om hierdie vergelyking te maak, wat getoon het dat die ATP konsentrasies homeostaties gebuffer was by verskillende aanbod tempos. Ek het ook hierdie aanpak gebruik om die bestendige-toestand gedrag in modelle van gisglikolise te vergelyk, en het bevind dat hulle nie noodwendig geoptimiseer is om ?n homeostatiese balans in die ATP konsentrasie te handhaaf nie, en dat in modelle vir hierdie organisme, die tempo van ATP produksie dikwels bepaal word deur die aanbod reaksies van glikoliese. Ek het verder ook bepaal of so ?n kinetiese model nuwe soorte gedrag kan beskryf, as dit aangepas word aan omstandighede wat verskil van dié waarvoor die model oorspronklik gekonstrueer was. Meer spesifiek, deur ?n model van bestendige-toestand gisglikolise te gebruik, kon ek wys dat klein veranderinge aan die oorspronkline ensiem konsentrasies genoeg was om ?n ossilerende model te verkry, wat opmerklik ooreenstem met die eksperimenteel waargenome ossilasies. Let ook daarop dat sommige van hierdie ensiem konsentrasie veranderinge plaasvind tydens die voorafbehandeling van die selle, wat essensieel is om die ossilasies waar te neem. Tot die beste van my kennis is die model wat ek met hierdie prosedures verkry het, die eerste gedetaileerde kinetiese model wat die eksperimenteel waargenome sterk sinkronisasie in ossilerende gis populasies voorspel. Om gis glikolitiese modelle te vergelyk in terme van hul dinamiese gedrag, het ek ?n raamwerk wat in fisika en ingeneurswese gebruik word, ingespan om ?n vektor-gebasseerde, twee dimensionele grafiese voorstelling van die ossilerende metaboliete en reaksies te maak. Hierdie raamwerk het dit nie net moontlik gemaak om ?n kompakte vergelyking van ?n stel modelle te maak nie, maar ek kon ook die bydrae van interaksies in die netwerk tot transduksie van die ossilasies kwantifiseer. Ek kon verder onderskeid tref tussen die verskillende ossilasiemeganismes vir elk van die modelle, en het ook gedemonstreer hoe die raamwerk gebruik kan word om sulke voorstellings vir eksperimentele datastelle te skep.

Response of flexible structures — such as cable-supported bridges and aircraft wings — is associated with a number of uncertainties in structural and flow parameters. This thesis is aimed at efficient uncertainty quantification in a few such flow and flow-induced structural response problems. First, the uncertainty quantification in the lift force exerted on a submerged body in a potential flow is considered. To this end, a new method — termed here as semi-intrusive stochastic perturbation (SISP) — is proposed. A sensitivity analysis is also performed, where for the global sensitivity analysis (GSA) the Sobol’ indices are used. The polynomial chaos expansion (PCE) is used for estimating these indices. Next, two stability problems —divergence and flutter — in the aeroelasticity are studied in the context of reliability based design optimization (RBDO). Two modifications are proposed to an existing PCE-based metamodel to reduce the computational cost, where the chaos coefficients are estimated using Gauss quadrature to gain computational speed and GSA is used to create nonuniform grid to reduce the cost even further. The proposed method is applied on a rectangular unswept cantilever wing model. Next, reliability computation in limit cycle oscillations (LCOs) is considered. While the metamodel performs poorly in this case due to bimodality in the distribution, a new simulation-based scheme proposed to this end. Accordingly, first a reduced-order model (ROM) is used to identify the critical region in the random parameter space. Then the full-scale expensive model is run only over a this critical region. This is applied to the rectangular unswept cantilever wing with cubic and fifth order stiffness terms in its equation of motion. Next, the wind speed is modeled as a spatio-temporal process, and accordingly new representations of spatio-temporal random processes are proposed based on tensor decompositions of the covariance kernel. These are applied to three problems: a heat equation, a vibration, and a readily available covariance model for wind speed. Finally, to assimilate available field measurement data on wind speed and to predict based on this assimilation, a new framework based on the tensor decompositions is proposed. The framework is successfully applied to a set of measured data on wind speed in Ireland, where the prediction based on simulation is found to be consistent with the observed data.