This paper presents an efficient methodology to build a modal solution emulator for the probabilistic study of geometrically mistuned bladed rotors by using the newly developed localized-Galerkin multifidelity (LGMF) modeling and eigensolution reanalysis (ER) with the symmetric successive matrix inversion (SSMI) methods. The key idea of the mistuned blade emulator is to establish a reduced functional relationship between the stochastic geometric variations and the disturbed modal responses. The prediction accuracy of an emulator generally depends on how many training samples of modal solutions are available and how well the potential modal switching due to stochastic mistuning is captured. To reduce the computational costs of generating training samples without sacrificing accuracy, this paper introduces the collaborative framework of the new approaches of multifidelity (MF) modeling and ER. The proposed framework is demonstrated for its computational benefits with several numerical examples including the point-cloud scanned mistuned blade problem.

The mistuning problem of quasi periodic structures is subject of numerous scientific investigations for more than 50 years. Researchers developed reduced order models to reduce the computational costs of mistuning investigations including finite element models. One question which has also high practical relevance is the identification of mistuning based on modal properties. In the present work, a new identification method based on the Subset of Nominal System Modes method (SNM) is presented. The input data for the identification procedure consists solely of the mistuned natural frequencies of the investigated mode family as well as of the corresponding mistuned mode shapes in the form of one degree of freedom per sector. The reduction basis consists of the tuned mode shapes of the investigated mode family. Furthermore, the proposed identification method allows for the inclusion of centrifugal effects like stress stiffening and spin softening without additional computational effort. From this point of view the presented method is also appropriate to handle centrifugal effects in reduced order models using a minimum set of input data compared to existing methods. The powerfulness of the new identification method is demonstrated on the example of an axial compressor blisk. Finite element calculations including geometrical mistuning provide the data base for the identification procedure. The correct functioning of the identification method including measurement noise is also validated to show the applicability to a case of application where real measurement data is available.

Integrally bladed rotors (IBRs) are meant to be rotationally periodic structures. However, unique variations in geometries and material properties from sector-to-sector, called mistuning, destroy the structural periodicity. This results in mode localization that can induce forced response levels greater than what is predicted with a tuned analysis. Furthermore, mistuning and mode localization are random processes that require stochastic treatments when analyzing the distribution of fleet responses. Generating this distribution can be computationally intensive when using the full finite element model (FEM). To overcome this expense, reduced-order models (ROMs) have been developed to accommodate fast calculations of mistuned forced response levels for a fleet of random IBRs. Usually, ROMs can be classified by two main families: frequency-based and geometry-based methods. Frequency-based ROMs assume mode shapes do not change due to mistuning. However, this assumption has been shown to cause errors that propagate to the fleet distribution. To circumvent these errors, geometry-based ROMs have been developed to provide accurate predictions. However, these methods require recalculating modal data during ROM formulations. This increases the computational expense in computing fleet distributions. A new geometry-based ROM is presented to reduce this cost. The developed ROM utilizes a Bayesian surrogate model in place of sector modal calculations required in ROM formulations. The method, surrogate modal analysis for geometry mistuning assessments (SMAGMA), will propagate uncertainties of the surrogate prediction to forced response. ROM accuracies are compared to the true forced response levels and results computed by a frequency-based ROM.

Can-annular combustors consist of $N$ distinct cans set up symmetrically around the axis of the gas turbine. Each can is connected to the turbine inlet by means of a transition duct. At the turbine inlet a small gap between the neighbouring transition ducts allows acoustic communication between the cans. Thermoacoustic pulsations in the cans are driven by the respective flames, but also the communication between neighbouring cans through the gap plays a significant role. In this study we focus on the effect of the background noise intensity and of the nonlinear flame saturation. We predict how usually clusters of thermoacoustic modes are unstable in the linear regime and compete with each other in the nonlinear regime, each cluster consisting of axial, azimuthal and push-pull modes. Since linear theory cannot predict the nonlinear solution, stochastic simulations are run to study the nonlinear solution in a probabilistic sense. One outcome of these simulations are the various pulsation patterns, which are in principle different from one can to the next. We recover how a stronger flame response in one can can give rise to the phenomenon of mode localization, but also how the nonlinearity of the flame saturation and the competition between modes have an effect on the nonlinear mode shape. We finally predict the coherence and phase between cans on the linearized system subject to noise, and compare the predictions with engine measurements, in terms of spectra of amplitude in each can and coherence and phase, observing a good match.

In the structural dynamics design process of turbomachines, Coriolis effects are usually neglected. This assumption holds true if no pronounced interaction between the shaft and disk occurs or if the radial blade displacements are negligible. For classical rotordynamic investigations or for machines where the disk is comparatively thin or weak, Coriolis effects as well as centrifugal effects like stress stiffening and spin softening have to be taken into account. For the analysis of complex structures, the finite element method is today the most commonly used modeling approach. To handle the numerical effort in such an analysis, the aim of the present work is the further development of an existing reduced order model, which also allows the consideration of Coriolis effects without the loss of accuracy for a wide range of rotational speeds. In addition to the investigation of the tuned design of the bladed disk using cyclic boundary conditions, the described method is also appropriate to investigate mistuning phenomena including Coriolis effects. Due to the fact that the computation time can be reduced by two orders of magnitude, the method also opens up the possibility for performing probabilistic mistuning investigations including Coriolis effects.

A new method has been developed for sensitivity calculations of modal characteristics of bladed disks made of anisotropic materials. The method allows the determination of the sensitivity of the natural frequencies and mode shapes of mistuned bladed disks with respect to anisotropy angles that define the crystal orientation of the monocrystalline blades using full-scale finite element models. An enhanced method is proposed to provide high accuracy for the sensitivity analysis of mode shapes. An approach has also been developed for transforming the modal sensitivities to coordinate systems (CS) used in industry for description of the blade anisotropy orientations. The capabilities of the developed methods are demonstrated on examples of a single blade and a mistuned realistic bladed disk finite element models. The modal sensitivity of mistuned bladed disks to anisotropic material orientation is thoroughly studied.

It is often desirable to identify the critical components that are active in a particular mode shape or an operational deflected shape (ODS) in a complex rotordynamic system with multiple rotating groups and bearings. The energy distributions can help identify the critical components of a rotor bearing system that may be modified to match the design requirements. Although the energy expressions have been studied by researchers in the past under specific limited conditions, these expressions require computing the displacements and velocities of all degrees-of-freedom (DOFs) over one full cycle. They do not address the overall time dependency of the energies and energy distributions, and their effect on the interpretation of a mode shape or an ODS. Moreover, a detailed finite element formulation of these energy expressions including the effects of anisotropy, skew-symmetric stiffness, viscous and structural damping have not been identified by the authors in the open literature. In this article, a detailed account of orbit characteristics and planarity for isotropic and anisotropic systems is presented. The effect of orbit characteristics on the energy expressions is then discussed. An elegant approach to obtaining time-dependent kinetic and strain energies of a mode shape or an ODS directly from the structural matrices and complex eigenvectors/displacement vectors is presented. The expressions for energy contributed per cycle by various types of damping and the destabilizing skew-symmetric stiffness that can be obtained in a similar way are also shown. The conditions under which the energies and energy distributions are time-invariant are discussed. An alternative set of energy expressions for isotropic systems with the DOFs reduced by half is also presented.

The potential of intentional mistuning to reduce the maximum forced response is analyzed within the development of an axial turbine blisk for ship diesel engine turbocharger applications. The basic idea of the approach is to provide an increased aerodynamic damping level for particular engine order (EO) excitations and mode shapes without any significant distortions of the aerodynamic performance. The mistuning pattern intended to yield a mitigation of the forced response is derived from an optimization study applying genetic algorithms. Two blisk prototypes have been manufactured a first one with and another one without employing intentional mistuning. Hence, the differences regarding the real mistuning and other modal properties can be experimentally determined and evaluated as well. In addition, the experimental data basis allows for updating structural models which are well suited to compute the forced response under operational conditions. In this way, the real benefit achieved with the application of intentional mistuning is demonstrated.

A novel methodology for linear stability analysis of high-frequency thermoacoustic oscillations in gas turbine combustors is presented. The methodology is based on the linearized Euler equations (LEEs), which yield a high-fidelity description of acoustic wave propagation and damping in complex, nonuniform, reactive mean flow environments, such as encountered in gas turbine combustion chambers. Specifically, this work introduces three novelties to the community: (1) linear stability analysis on the basis of linearized Euler equations. (2) Explicit consideration of three-dimensional, acoustic oscillations at screech level frequencies, particularly the first-transversal mode. (3) Handling of noncompact flame coupling with LEE, that is, the spatially varying coupling dynamics between perturbation and unsteady flame response due to small acoustic wavelengths. Two different configurations of an experimental model combustor in terms of thermal power and mass flow rates are subject of the analysis. Linear flame driving is modeled by prescribing the unsteady heat release source term of the linearized Euler equations by local flame transfer functions, which are retrieved from first principles. The required steady-state flow field is numerically obtained via computational fluid dynamics (CFD), which is based on an extended flamelet-generated manifold (FGM) combustion model, taking into account heat transfer to the environment. The model is therefore highly suitable for such types of combustors. The configurations are simulated, and thermoacoustically characterized in terms of eigenfrequencies and growth rates associated with the first-transversal mode. The findings are validated against experimentally observed thermoacoustic stability characteristics. On the basis of the results, new insights into the acoustic field are discussed.

In operation, rotating bladed disks (blisks) are often subject to high levels of dynamic loading, resulting in large amplitudes of forced vibrations especially at resonance. Moreover, variations in structural properties of individual sectors, referred to as mistuning, can lead to strain energy localization and can amplify forced responses. To prevent damages caused by high cycle fatigue, various frictional damping sources are introduced to dissipate vibration energy. Due to the nonlinear behavior of frictional contacts, conventional methods to study the dynamics of the blisk–damper systems are based often on numerical time integration, which is time-consuming and can be computationally prohibitive due to the large sizes of commercial blisk models. Existing techniques for model reduction either rely heavily on cyclic symmetry of the blisk–damper system or are based on component mode synthesis (CMS). However, in the presence of mistuning, cyclic symmetry no longer exists. Also, mistuning is random and best studied statistically. Repetitive CMS condensation for a large amount of random mistuning patterns can lead to a computationally formidable task. This paper presents a reduced-order modeling (ROM) technique to efficiently capture the nonlinear dynamic responses of blisk–damper systems with both small perturbations in blade material properties (small mistuning) and significant changes in the blisk geometries (large mistuning). The ROMs are formed by projecting the blisk–damper systems onto a novel mode basis that mimics the contact behavior. This mode basis contains normal mode shapes of the mistuned blisk–damper systems with either sliding or sticking conditions enforced on the contact surfaces. These mode shapes are computed through the N-PRIME method, a technique recently developed by the authors to efficiently obtain mode shapes for blisks with simultaneous large and small mistuning. The resulting modal nonlinear equations of motion (EOM) are solved by a hybrid frequency/time (HFT) domain method with continuation. In the HFT method, the contact status and friction forces are determined in the time domain by a quasi-two-dimensional contact model at each contact point, whereas the modal EOM are solved in the frequency domain according to a harmonic balance formulation. The forced responses computed by the proposed ROMs are validated for two systems with distinct mistuning patterns. A statistical analysis is performed to study the effectiveness of the frictional dampers under random mistuning patterns.

It is well known that the vibrational behavior of a mistuned bladed disk differs strongly from that of a tuned bladed disk. A large number of publications dealing with the dynamics of mistuned bladed disks are available in the literature. The vibrational phenomena analyzed in these publications are either forced vibrations or self-excited flutter vibrations. Nearly, all published literature on the forced vibrations of mistuned blades disks considers harmonic, i.e., steady-state, vibrations, whereas the self-excited flutter vibrations are analyzed by the evaluation of the margin against instabilities by means of a modal, or rather than eigenvalue, analysis. The transient forced response of mistuned bladed disk is not analyzed in detail so far. In this paper, a computationally efficient mechanical model of a mistuned bladed disk to compute the transient forced response is presented. This model is based on the well-known fundamental model of mistuning (FMM). With this model, the statistics of the transient forced response of a mistuned bladed disk is analyzed and compared to the results of harmonic forced response analysis.

This paper is focused on the analysis of effects of mistuning on the forced response of gas turbine engine bladed disks vibrating in the frequency ranges corresponding to higher modes. For high modes considered here, the blade aerofoils are deformed during vibrations and the blade mode shapes differ significantly from beam mode shapes. A model reduction technique is developed for the computationally efficient and accurate analysis of forced response for bladed disks vibrating in high-frequency ranges. The high-fidelity finite element (FE) model of a tuned bladed disk sector is used to provide primary information about dynamic properties of a bladed disk, and the blade mistuning is modeled by specially defined mistuning matrices. The forced response displacement and stress amplitude levels are studied. The effects of different types of mistuning are examined, and the existence of high amplifications of mistuned forced response levels is shown for high-mode vibrations: in some cases, the resonance peak response of a tuned structure can be lower than out-of-resonance amplitudes of its mistuned counterpart.

Modified modal domain analysis (MMDA), a reduced order modeling technique, is applied to a geometrically mistuned integrally bladed rotor to obtain its natural frequencies, mode shapes, and forced response. The geometric mistuning of blades is described in terms of proper orthogonal decomposition (POD) of the coordinate measurement machine (CMM) data. Results from MMDA are compared to those from the full (360 deg) rotor Ansys model. It is found that the MMDA can accurately predict natural frequencies, mode shapes, and forced response. The effects of the number of POD features and the number of tuned modes used as bases for model reduction are examined. Results from frequency mistuning approaches, fundamental mistuning model (FMM) and subset of nominal modes (SNM), are also generated and compared to those from full (360 deg) rotor Ansys model. It is clearly seen that FMM and SNM are unable to yield accurate results whereas MMDA yields highly accurate results.

Traditional experimental modal testing techniques rely on controlled and measured excitation together with measured responses in order to identify the mode shape, natural frequency, and damping factor of each mode. Applying a controlled and measured excitation to a rotor train when in operation is logistically difficult and especially challenging in the field. Operational modal analysis (OMA) identifies the modal parameters of a system from measurement of response due to some (unknown) excitation. OMA has proven successful over the past several decades on nonrotating structures but has relatively rarely been applied to rotating machinery. Case studies are presented demonstrating the use of OMA in identifying torsional modes on an electric motor driven reciprocating compressor, on a diesel engine driven fire water pump, and on a marine propulsion system. In contrast to lateral modes, torsional modes of rotor trains are typically not speed dependent. However, phenomena exist whereby the torsional modes may be different at stand still, off-load and at different loads. The case studies provide examples of such phenomena and also of significant differences between predicted and measured behavior which suggests that improvements in industrial practice would be beneficial. Such improvements should be based on reconciliation of measured and predicted behavior and OMA offers a valuable tool to facilitate this. OMA provides a significant benefit in investigating and understanding torsional behavior in operation.

Successful, efficient turbine design requires a thorough understanding of the underlying physical phenomena. This paper investigates the flutter phenomenon of low pressure turbine (LPT) blades seen in aircraft engines and power turbines. Computational fluid dynamics (CFD) analysis will be conducted in a two-dimensional (2D) sense using a frequency domain Reynolds-averaged Navier—Stokes (RANS) solver on a publicly available LPT airfoil geometry: École Polytechnique Fédérale De Lausanne (EPFL's) Standard Configuration 4. An emphasis is placed on revealing the underlying physics behind the threatening LPT flutter mechanism. To this end, flutter sensitivity analysis is conducted on three key parameters: reduced frequency, mode shape, and Mach number. Additionally, exact 2D acoustic resonance interblade phase angles (IBPAs) are analytically predicted as a function of reduced frequency. Made evident via damping versus IBPA plots, the CFD model successfully captures the theoretical acoustic resonance predictions. Studies of the decay of unsteady aerodynamic influence coefficients away from a reference blade are also presented. The influence coefficients provide key insights to the harmonic content of the unsteady pressure field. Finally, this work explores methods of normalizing the work per cycle by the exit dynamic pressure.

This paper deals with further development of modified modal domain analysis (MMDA), which is a breakthrough method in the reduced order modeling of a bladed rotor with geometric mistuning. The main focus of this paper is to show that deviations in mass and stiffness matrices due to mistuning, estimated by Taylor series expansions in terms of independent proper orthogonal decomposition variables representing geometric variations of blades, can be used for MMDA. This result has rendered Monte Carlo simulation of the response of a bladed rotor with geometric mistuning to be easy and computationally efficient.

A new effective method for comprehensive modeling of gas flow effects on vibration of nonlinear vibration of bladed disks has been developed for a case when the effect of the gas flow on the mode shapes is significant. The method separates completely the structural dynamics calculations from the significantly more computationally expensive computational fluid dynamics (CFD) calculations while providing the high accuracy of modeling for aerodynamic effects. A comprehensive analysis of the forced response using the new method has been performed for a realistic turbine bladed disk with root-disk joints, tip, and under-platform dampers. The full chain of aerodynamic and structural calculations are performed: (i) determination of boundary conditions for CFD, (ii) CFD analysis, (iii) calculation of the aerodynamic characteristics required by the new method, and (iv) nonlinear forced response analysis using the modal aerodynamic influence matrix (MAIM). The efficiency of the friction damping devices has been studied and compared for several resonance frequencies and engine orders. Advantages of the method for aerodynamic effect modeling have been demonstrated.

This paper presents an experimentally driven model updating approach to address the dynamic inaccuracy of the nominal finite element (FE) rotor model of a machining spindle supported on active magnetic bearings. Modeling error is minimized through the application of a numerical optimization algorithm to adjust appropriately selected FE model parameters. Minimizing the error of both resonance and antiresonance frequencies simultaneously accounts for rotor natural frequencies as well as for their mode shapes. Antiresonance frequencies, which are shown to heavily influence the model’s dynamic properties, are commonly disregarded in structural modeling. Evaluation of the updated rotor model is performed through comparison of transfer functions measured at the cutting tool plane, which are independent of the experimental transfer function data used in model updating procedures. Final model validation is carried out with successful implementation of robust controller, which substantiates the effectiveness of the model updating methodology for model correction.

The compression ring-bore conjunction accounts for significant frictional parasitic losses relative to its size. The prerequisite to improving the tribological performance of this contact is a fundamental understanding of ring dynamics within the prevailing transient nature of the regime of lubrication. Studies reported thus far take into account ring-bore conformance based on static fitment of the ring within an out-of-round bore, whose out-of-circularity is affected by manufacturing processes, surface treatment, and assembly. The static fitment analyses presume quasi-static equilibrium between ring tension and gas pressure loading with generated conjunctional pressures. This is an implicit assumption of ring rigidity while in situ. The current analysis considers the global modal behavior of the ring as an eigenvalue problem, thus including its dynamic in-plane behavior in the tribological study of a mixed-hydrodynamic regime of lubrication. The results show that the contact transit time is shorter than that required for the ring to reach steady state condition. Hence, the conjunction is not only subject to transience on account of changing contact kinematics and varied combustion loading, but also subject to perpetual ring transient dynamics. This renders the ring-bore friction a more complex problem than usually assumed in idealized ring fitment analyses. An interesting finding of the analysis is increased ring-bore clearance at and in the vicinity of top dead center, which reduces the ring-sealing effect and suggests a possible increase in blow-by. The current analysis, integrating ring in-plane modal dynamics and mixed regime of lubrication, includes salient features, which are a closer representation of practice, an approach which has not hitherto been reported in literature.

A method has been developed for high-accuracy analysis of forced response levels for mistuned bladed disks vibrating in gas flow. Aerodynamic damping, the interaction of vibrating blades through gas flow, and the effects of structural and aerodynamic mistuning are included in the bladed disk model. The method is applicable to cases of high mechanical coupling of blade vibration through a flexible disk and, possibly shrouds, to cases with stiff disks and low mechanical coupling. The interaction of different families of bladed disk modes is included in the analysis providing the capability of analyzing bladed disks with pronounced frequency veering effects. The method allows the use of industrial-size sector models of bladed disks for analysis of forced response of a mistuned structure. The frequency response function matrix of a structurally mistuned bladed disk is derived with aerodynamic forces included. A new phenomenon of reducing bladed disk forced response by mistuning to levels that are several times lower than those of their tuned counterparts is revealed and explained.