Bladed disks are critical structural components in jet engines and other turbomachinery. The nominal design for a bladed disk is typically assumed to have identical blades. However, there are always small, random variations in the blade properties due to manufacturing tolerances, material defects, and operational wear. These blade-to-blade discrepancies, called mistuning, can have a dramatic effect on bladed disk vibration. In particular, mistuning can cause localization of the response in a small region of the bladed disk, leading to higher blade stress and high-cycle fatigue concerns. While comprehensive analytical and computational studies of mistuning have been performed, relatively few experimental investigations have been conducted. The primary objective of this research is to experimentally investigate the fundamental structural dynamics of mistuned bladed disks, and to achieve a physical understanding of mistuning effects by accounting for the influence of important phenomena that have been largely neglected in previous mistuning models and system identification algorithms. First, a systematic experimental approach is presented to validate a new mistuning identification and model updating algorithm for single-piece bladed disks, or blisks. It is shown that only a few system response measurements taken at resonant frequencies are required to identify the blade stiffness mistuning parameters and the model updating parameters referred to as cyclic modeling error. By incorporating a model updating procedure, the accuracy of the mistuning identification results are significantly improved. Second, an alternative approach for vibration testing of many mistuning patterns is proposed and validated. In particular, varying the external forcing function provided to the blades is used to mimic the influence of structural blade property mistuning on the vibration response. Since it is much easier and more efficient to vary the external excitation than to physically alter the blades, this work opens the possibility of running an experimental analogue of a Monte Carlo simulation. Finally, the mistuning identification method is extended to also identify the forcing amplitude and phase applied to each blade. This approach shows promise as a powerful tool for accelerating calibration procedures, as well as for improving the accuracy and capability of experimental methods for bladed disks.

Les disques aubagés, que l’on trouve dans les turbomachines, sont des structures complexes dont le comportement vibratoire est généralement déterminé par l’exploitation de conditions de symétrie dans leur configuration nominale. Cette symétrie disparaît lorsque l’on assemble plusieurs de ces disques pour former un rotor ou que l’on introduit une variabilité spatiale des paramètres mécaniques (on parle de désaccordage intentionnel ou non). Le raffinement des maillages, nécessaire à une évaluation correcte de la répartition des contraintes, conduirait à des modèles de rotor complet de taille prohibitive (plusieurs dizaines de millions de degrés de liberté). L’objectif de cette thèse est donc l’introduction de méthodologies de réduction qui par combinaison de calculs acceptables permettent d’étudier de façon fine la dynamique d’ensemble sur des modèles 3D fins multi-étages et potentiellement désaccordés. L’étude des transformations de Fourier séparées des réponses de chaque étage permet, dans un premier temps, de bien comprendre les effets de couplage inter-harmonique liés au couplage inter-disque et au désaccordage. A partir de ce constat, une première méthode utilise les résultats de calculs en symétrie cyclique et à secteur encastré pour construire un modèle de secteur exact pour certains modes dits cibles et de très bonne qualité pour les autres modes. Cette méthode est ensuite étendue au cas multi-étage en construisant des bases de réduction de secteur par combinaison de solutions mono-harmoniques. Les illustrations montrent que la méthodologie proposée permet le traitement de modèles de très grande taille, tout en restant compatible avec une grande richesse de post-traitements (calculs de modes, calculs de réponses forcées, analyses de leur contenu harmonique spatial, répartition d’énergie et effets de localisation...). La méthodologie est enfin étendue à la gestion de modèles paramétrés en vitesse de rotation. L’enrichissement des ensembles de modes cibles par des calculs à trois vitesses permet ainsi une reconstruction rapide de l’évolution des fréquences pour l’ensemble d’un intervalle. / Bladed disks found in turbomachines are complex structures whose vibration characteristics are generally determined by exploiting the symmetry properties of their nominal configuration. This symmetry no longer exists either when disks are assembled to form a rotor or when discrepancies in the mechanical parameters are introduced (intentional or unintentional mistuning). Fine meshes required to correctly evaluate stress distributions would lead to prohibitive model sizes (typically a few million degrees of freedom). The objective of this thesis is to introduce model reduction techniques that rely on the combination of separate computations of acceptable size. This provides a means for in-depth studies of the behaviour of dense 3D models of multi-stage bladed rotors with possible mistuning. At first, Fourier transforms performed separately on each individual disk allows to understand the inter-harmonic coupling induced by inter-stage coupling and mistuning. From this study, a first method uses cyclically symmetric solutions plus sector modes with fixed inter-sector interfaces to build a reduced sector model. The latter is exact for target modes and very accurate for others. This method is extended to multi-stage assemblies by employing multi-stage mono-harmonic eigensolutions. Illustrations focus on the proposed methodology that enables to deal with large scale industrial models while remaining compatible with various post-processing procedures (free or forced response computations, analysis of their spatial harmonic content, energy distributions and localization effects...). This methodology is finally extended to the handling of parametric models depending on the rotation speed. The enrichment of the initial sets of target vectors with computations at three rotation speeds enables a fast and accurate recovery of the evolution of the eigenfrequencies with respect to the rotation speed in any operating range.

Two state of the art finite element reduction techniquespreviously validated against the direct finite element method,one based on classical modal analysis and another based oncomponent mode synthesis, are applied for efficient mistunedfree vibration and forced response analysis of several bladeddisk geometries. The methods are first applied to two testcases in order to demonstrate the differences in computationalefficiency as well as to validate the methods againstexperimental data. As previous studies have indicated, nonoticeable differences in accuracy are detected for the currentapplications, while the method based on classical modalanalysis is significantly more efficient. Experimental data(mistuned frequencies and mode shapes) available for one of thetwo test cases are compared with numerical predictions, and agood match is obtained, which adds to the previous validationof the methods (against the direct finite element method). The influence of blade-to-blade coupling and rotation speedon the sensitivity of bladed disks to mistuning is thenstudied. A transonic fan is considered with part span shroudsand without shrouds, respectively, constituting a high and alow blade-to-blade coupling case. For both cases, computationsare performed at rest as well as at various rotation speeds.Mistuning sensitivity is modelled as the dependence ofamplitude magnification on the standard deviation of bladestiffnesses. The finite element reduction technique based onclassical modal analysis is employed for the structuralanalysis. This reduced order model is solved for sets of randomblade stiffnesses with various standard deviations, i.e. MonteCarlo simulations. In order to reduce the sample size, thestatistical data is fitted to a Weibull (type III) parametermodel. Three different parameter estimation techniques areapplied and compared. The key role of blade-to-blade coupling,as well as the ratio of mistuning to coupling, is demonstratedfor the two cases. It is observed that mistuning sensitivityvaries significantly with rotation speed for both fans due toan associated variation in blade-to-blade coupling strength.Focusing on the effect of one specific engine order on themistuned response of the first bending modes, it is observedthat the mistuning sensitivity behaviour of the fan withoutshrouds is unaffected by rotation at its resonant condition,due to insignificant changes in coupling strength at thisspeed. The fan with shrouds, on the other hand, shows asignificantly different behaviour at rest and resonant speed,due to increased coupling under rotation. Comparing the twocases at resonant rotor speeds, the fan without shrouds is lessor equally sensitive to mistuning than the fan with shrouds inthe entire range of mistuning strengths considered. This thesis’scientific contribution centres on themistuning sensitivity study, where the effects of shrouds androtation speed are quantified for realistic bladed diskgeometries. However, also the validation of two finite elementreduction techniques against experimental measurementsconstitutes an important contribution. / NR 20140805

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Methods of determining natural frequencies of the D76D88, B76D88, A86E93, C86G90, C86L90 and C125L89 turbine wheel designs for various environmental conditions were investigated by application of Finite Element Analysis and beam theory. Modelling and simulation methods were developed ; the first method composed of 15 finite element simulations ; the second composed of 15 finite element simulations and a set of experimental frequency survey results; the third composed of 5 simulations , an incorporated mathematical model and a set of experimental frequency survey results. Each of these methods was designed to allow prediction of resonant frequency changes across a range of exhaust gas temperature and shaft rotational speed. For the new modelling and simulation methods, an analysis template and a plotting tool were developed using Microsoft Excel and MATLAB software. A graph showing a frequency-temperature-speed variations and a Campbell Diagram that incorporates material stiffening and softening effects across a range of rotational speeds was designed, and applied to the D76D88, B76D88, A86E93, C86G90, C86L90 and C125L89 turbine wheel designs. New design methodologies for turbine wheels were formulated and validated, showing a good agreement with a range of data points from frequency survey, strain-gauge telemetry and laser tip-timing test results. The results from the new design method were compared with existing single compensation factor methodology, and showed a great improvement in accuracy of prediction of modal vibration. A new nomenclature for the mode shapes of a turbocharger’s blade was proposed, designed and demonstrated to allow direct identification of associated mode shape. It is concluded that Finite Element Analysis combined with the frequency survey is capable of predicting changes in turbine natural frequencies and, when incorporated into the existing turbine design methodology, resulted in a major improvement in the accuracy of the predictions of vibration frequency.

ABSTRACT POSSIBILISTIC INTERPRETATION OF MISTUNING IN BLADED DISKS BY FUZZY ALGEBRA Karatas, Hamit &Ccedil / aglar M.S., Department of Mechanical Engineering Supervisor: Prof. Dr. H. Nevzat &Ouml / zg&uuml / ven Co-supervisor: Asst. Prof. Dr. Ender Cigeroglu September 2012, 103 pages This study aims to define the possibilistic interpretation of mistuning and examine the way of determining the worst case situations and assessing reliability value to that case by using possibilistic methods. Furthermore, in this study, benefits of using possibilistic interpretation of mistuning in comparison to probabilistic interpretation of mistuning are investigated. For the possibilistic analysis of mistuned structures, uncertain mistuning parameters are modeled as fuzzy variables possessing possibility distributions. In this study, alpha-cut representations of fuzzy numbers are used which makes fuzzy variables to be represented by interval numbers at each and every confidence level. The solution of fuzzy equations of motion is governed by fuzzy algebra methods. The bounds of the solution of the fuzzy equation of motion, i.e. fuzzy vibration responses of the mistuned structure, are determined by the extension principle of fuzzy functions. The performance of the method for possibilistic interpretation of mistuning is investigated by comparing it to the probabilistic methods both computational and accuracy wise. For the comparison study, two different optimization tools &ndash / genetic algorithm as the global optimization tool and constrained nonlinear minimization method as the gradient based optimization tool- are utilized in possibilistic analysis and they are compared to solutions of probabilistic methods resulted from Monte-Carlo method. The performances of all of the methods are tested on both a cyclically symmetric lumped parameter model and a realistic reduced order finite element model.

This study aims to define the possibilistic interpretation of mistuning and examine the way of determining the worst case situations and assessing reliability value to that case by using possibilistic methods. Furthermore, in this study, benefits of using possibilistic interpretation of mistuning in comparison to probabilistic interpretation of mistuning are investigated. For the possibilistic analysis of mistuned structures, uncertain mistuning parameters are modeled as fuzzy variables possessing possibility distributions. In this study, alpha-cut representations of fuzzy numbers are used which makes fuzzy variables to be represented by interval numbers at each and every confidence level. The solution of fuzzy equations of motion is governed by fuzzy algebra methods. The bounds of the solution of the fuzzy equation of motion, i.e. fuzzy vibration responses of the mistuned structure, are determined by the extension principle of fuzzy functions. The performance of the method for possibilistic interpretation of mistuning is investigated by comparing it to the probabilistic methods both computational and accuracy wise. For the comparison study, two different optimization tools &ndash / genetic algorithm as the global optimization tool and constrained nonlinear minimization method as the gradient based optimization tool- are utilized in possibilistic analysis and they are compared to solutions of probabilistic methods resulted from Monte-Carlo method. The performances of all of the methods are tested on both a cyclically symmetric lumped parameter model and a realistic reduced order finite element model.

High cycle fatigue (HCF) failure has been studied extensively over the last two decades. Its impact on jet engines is severe enough that may result in engine losses and even life losses. The main requirement for fatigue life predictions is the stress caused by mechanical vibrations. One of the factors which have major impact on the vibratory stresses of bladed disk assemblies is a phenomenon called &ldquo / mistuning&rdquo / which is defined as the vibration localization caused by the loss of cyclic periodicity which is a consequence of inter&amp / #8208 / blade variations in structural properties. In this thesis, component mode synthesis method (CMSM) is combined with nonlinear forced response analysis in modal domain. Newton&amp / #8208 / Raphson and arc length continuation procedures are implemented for the solution. The component mode synthesis method introduces the capability of imposing mistuning on the modal properties of each blade in the assembly. Forced response analysis in modal domain reduces the problem size via mode truncation. The main advantage of the proposed method is that it is capable of calculating nonlinear forced response for all the degrees&amp / #8208 / of&amp / #8208 / freedom at each blade with less computational effort. This makes it possible to make a stress analysis at resonance conditions. The case studies presented in this thesis emphasize the importance of number of modes retained in the reduced order model for both CMSM and nonlinear forced response analysis. Furthermore, the results of the case studies have shown that both nonlinearity and mistuning can cause shifts in resonance frequencies and changes in resonance amplitudes. Despite the changes in resonance conditions, the shape of the blade motion may not be affected.

Methods of determining natural frequencies of the D76D88, B76D88, A86E93, C86G90, C86L90 and C125L89 turbine wheel designs for various environmental conditions were investigated by application of Finite Element Analysis and beam theory. Modelling and simulation methods were developed ; the first method composed of 15 finite element simulations ; the second composed of 15 finite element simulations and a set of experimental frequency survey results; the third composed of 5 simulations , an incorporated mathematical model and a set of experimental frequency survey results. Each of these methods was designed to allow prediction of resonant frequency changes across a range of exhaust gas temperature and shaft rotational speed. For the new modelling and simulation methods, an analysis template and a plotting tool were developed using Microsoft Excel and MATLAB software. A graph showing a frequency-temperature-speed variations and a Campbell Diagram that incorporates material stiffening and softening effects across a range of rotational speeds was designed, and applied to the D76D88, B76D88, A86E93, C86G90, C86L90 and C125L89 turbine wheel designs. New design methodologies for turbine wheels were formulated and validated, showing a good agreement with a range of data points from frequency survey, strain-gauge telemetry and laser tip-timing test results. The results from the new design method were compared with existing single compensation factor methodology, and showed a great improvement in accuracy of prediction of modal vibration. A new nomenclature for the mode shapes of a turbocharger’s blade was proposed, designed and demonstrated to allow direct identification of associated mode shape. It is concluded that Finite Element Analysis combined with the frequency survey is capable of predicting changes in turbine natural frequencies and, when incorporated into the existing turbine design methodology, resulted in a major improvement in the accuracy of the predictions of vibration frequency. / Additional data files have been restricted by request.