Large Mistuning Research Articles

Investigation of Structurally and Aerodynamically Mistuned Oscillating Cascade Using Fully Coupled Method

Abstract There seems to be a lack of clear and systematic understanding of physical behavior and mechanisms of mistuned blade rows, particularly in the context of the aerodynamic mistuning versus the structural (frequency) mistuning. A high-fidelity fully coupled method is desirable to investigate the vibration characteristics of aeroelasticity problems with strong fluid–structure interaction effects, as well as blade mistuning effects. In this work, the direct nonlinear time-domain fully coupled method is adopted to investigate the dynamics mechanism of a mistuned oscillating cascade. The main objectives are two-folds, first to elucidate the basic vibration characteristics of a mistuned blade row, and second to examine the aeroelastic effects of mistuning. Three conditions of interest are considered: (a) the structural mistuning only, (b) the aerodynamic mistuning only, and (c) a combination of the two. The present results show that first a mistuned configuration tends to vibrate with the same frequency and a predominantly constant interblade phase angle. Vibration amplitudes of the blades vary significantly with a strong mode localization effect for the structural mistuning. For the concurrent structural-aerodynamic mistuning, the localization is stronger than in the standalone structural mistuning case. Second, a monotonic increase of the aeroelastic stability with the structural mistuning magnitude is observed. On the other hand, the aerodynamically mistuned cascade shows a stabilizing effect with a small amount of mistuning but exhibits a destabilizing effect with a large mistuning. Furthermore, these results indicate a quite remarkable interplay between the structural and the aerodynamic mistuning. At a condition where the tuned cascade is still stable, an aero-mistuning induced unstable behavior is observed. An introduction of a large magnitude of frequency mistuning which would be stabilizing for a tuned cascade, is instead shown to have a destabilizing effect for the present combined aero-structural mistuning case.

A Model Reduction Method for Bladed Disks With Large Geometric Mistuning Using a Partially Reduced Intermediate System Model

Abstract Reduced order models (ROMs) are widely used to enable efficient simulation of mistuned bladed disks. ROMs based on projecting the system dynamics into a subspace spanned by the modes of the tuned structure work well for small amounts of mistuning. When presented with large mistuning, including changes of geometry and number of finite element mesh nodes, advanced methods such as the pristine-rogue-interface modal expansion (PRIME) are necessary. PRIME builds a reduced model from two full cyclic symmetric analyses, one for the nominal and one for the modified type of sector. In this paper, a new reduced order model suitable for large mistuning with arbitrary mesh modifications is presented. It achieves higher accuracy than PRIME, while saving approximately 25% computational effort during the reduction process, when using the same number of cyclic modes. The new method gains its efficiency by recognizing that large modifications from damage or repair are unlikely to be exactly the same for multiple blades. It works by building a partially reduced intermediate model: All nominal sectors are reduced using cyclic modes of the tuned structure. The single modified sector is kept as the full model. For this reason, the new reduction method is called partially reduced intermediate system model (PRISM) method. The accuracy of the PRISM method is demonstrated on an axial compressor blisk and an academic blisk geometry.

Identification and influence factors analysis of blade crack mistuning in hard-coated blisk based on modified component mode mistuning reduced-order model

Blade crack will cause severe mistuning of hard-coated blisks, which will lead to vibration localization. To identify crack mistuning and analyze influence factors, in this study, a mistuning identification method of blade cracks in hard-coated blisks is presented based on modified component mode mistuning reduced-order model, in which the hard-coated blisk with blade crack is decomposed into a substructure of tuned hard-coated blisk and a substructure of coated blade with cracks. Crack mistuning of each coated blade can be obtained by a single identification calculation. After verifying the rationality of this identification method, the influence factors of blade crack mistuning are analyzed. The influence factors include the crack location on the coated blade (cracks occurring only in coating or only in blade substrate or both in blade substrate and coating), crack length, crack position in the radial direction of the blisk, and modal data type of coated blisk used for mistuning identification calculation. The research results show that, with the increase of crack length, the mistuning of crack occurring only in the coating does not increase continuously but decreases firstly and then increases. For the first bending modes, the closer the blade crack is to the blade root, the larger the mistuning is. For the second bending modes, the blade crack located at the position of maximum modal displacement will produce large mistuning. For hard-coated blisk with blade crack, these crack mistuning variation rules are of great significance to the dynamic analysis and the determination of the crack location.

Reduced Order Models for Largely Mistuned Blisks with Small Random Mistuning

Mistuning can break the cyclic symmetry of an ideally tuned blisk and cause its forced response to increase drastically. In the absence of cyclic symmetry, the finite element method (FEM) requires a vast amount of computational costs for predicting the dynamic characteristics of mistuned systems. To greatly reduce the computational costs of FEM, various reduced-order models (ROMs) have been developed. However, most existing ROMs are only effective for small mistuning or large mistuning situations. To resolve more complicated mistuning problems, this paper proposes two new approaches for blisks with both small mistuning and large mistuning. The first method extends the idea of the subset of nominal modes (SNM), in which modes of a largely mistuned system with small mistuning can be represented in terms of a subset of the pseudo-nominal system modes (SPNM). Because SPNM has to use system-level modes and calculation, the second method, called remaining damaged interface decomposition (HRDID), is further proposed, which is based on the basic principle of SPNM. It requires only sector-level modes and calculation. Furthermore, SPNM, HRDID, and FEM are used separately to calculate both the natural frequencies and mode shapes of mistuned academic blisks. The results of SPNM and HRDID are compared separately with those calculated by FEM to verify their accuracy. Finally, statistical analysis is performed to investigate the effect of random mistuning on a pseudo-nominal system.

A Statistical Characterization of the Effects and Interactions of Small and Large Mistuning on Multistage Bladed Disks

Abstract As turbomachinery systems continue to push the limits of modern technology, the modeling techniques being developed also continue to push modern computational limits. While modeling the pristine system during the design process is important, it is equally important to model behavior of mistuned systems, due to wear or manufacturing processes. This work carries out a statistical analysis on a two-stage system consisting of integrally bladed rotors. Various forms of large geometric mistuning, such as missing mass, bends, and dents, are considered. These forms of large mistuning are based on previously reported damage seen in actual engines that have ingested volcanic ash. This work also aims to investigate the interaction of these large mistuned systems with various levels of small mistuning to further understand this complex interaction. For every case studied, a reduced-order model (ROM) will be constructed that includes the effects of small and large mistuning. Large mistuning will only be applied to a single sector of a single stage. Random patterns of small mistuning will be applied to each case of large mistuning after which a modal analysis and a forced response analysis are run. By observing the energy distribution of each mode for the mistuned system, a qualitative trend can be created between various types and levels of large mistuning and the impact they have on changing the dynamics of the multistage system. The amplification factors from the forced response analyses help in understanding the impact small mistuning has when coupled with large mistuning and when the effects of small mistuning dominate over the large mistuning effects.

Response of large mistuned bladed disk assembly subjected to random excitations

In this study, the effects of large geometric mistuning on blade response are investigated considering the change of mistuning properties and excitation characteristics. The response variations because of large mistuning are negligible under white noise excitations and are highly significant under narrow band excitations. For the different excitation correlations, the responses are represented using non-dimensional variables in order for the results to be comparable to each other. It is shown that the ratio of the modal frequency between the tuned and large mistuned blades determines the propagation direction of the natural frequencies of the mistuned system, therefore, the responses of large mistuned blades depend primarily on their level of modal frequency.

Reduced-Order Modeling of Bladed Disks Considering Small Mistuning of the Disk Sectors

A model order reduction method based on the component mode synthesis for mistuned bladed disks is introduced, with one component for the disk and one component for each blade. The interface between the components at the blade roots is reduced using the wave-based substructuring (WBS) method, which employs tuned system modes. These system modes are calculated first, and used subsequently during the reduction of the individual components, which eliminates the need to build a partially reduced intermediate model with dense matrices. For the disk, a cyclic Craig–Bampton (CB) reduction is applied. The deviations of the stiffness and mass matrices of individual disk sectors are then projected into the cyclic basis of interior and interface modes of the disk substructure. Thereby, it is possible to model small disk mistuning in addition to large mistuning of the blades.

Multistage Blisk and Large Mistuning Modeling Using Fourier Constraint Modes and PRIME

Current efforts to model multistage turbomachinery systems rely on calculating independent constraint modes for each degree-of-freedom (DOF) on the boundary between stages. While this approach works, it is computationally expensive to calculate all the required constraint modes. This paper presents a new way to calculate a reduced set of constraint modes referred to as Fourier constraint modes (FCMs). These FCMs greatly reduce the number of computations required to construct a multistage reduced order model (ROM). The FCM method can also be integrated readily with the component mode mistuning (CMM) method to handle small mistuning and the pristine rogue interface modal expansion (PRIME) method to handle large and/or geometric mistuning. These methods all use sector-level models and calculations, which make them very efficient. This paper demonstrates the efficiency of the FCM method on a multistage system that is tuned and, for the first time, creates a multistage ROM with large mistuning using only sector-level quantities and calculations. The results of the multistage ROM for the tuned and large mistuning cases are compared with full finite element results and are found in good agreement.

Models for blisks with large blends and small mistuning

Small deviations of the structural properties of individual sectors of blisks, referred to as mistuning, can lead to localization of vibration energy and drastically increased forced responses. Similar phenomena are observed in blisks with large damages or repair blends. Such deviations are best studied statistically because they are random. In the absence of cyclic symmetry, the computational cost to predict the vibration behavior of blisks becomes prohibitively high. That has lead to the development of various reduced-order models (ROMs). Existing approaches are either for small mistuning, or are computationally expensive and thus not effective for statistical analysis. This paper discusses a reduced-order modeling method for blisks with both large and small mistuning, which requires low computational effort. This method utilizes the pristine, rogue and interface modal expansion (PRIME) method to model large blends. PRIME uses only sector-level cyclic modes strategically combined together to create a reduction basis which yields ROMs that efficiently and accurately model large mistuning. To model small mistuning, nodal energy weighted transformation (NEWT) is integrated with PRIME, resulting in N-PRIME, which requires only sector-level calculations to create a ROM which captures both small and large mistuning with minimized computational effort.The combined effects of large blends and small mistuning are studied using N-PRIME for a dual flow path system and for a conventional blisk. The accuracy of the N-PRIME method is validated against full-order finite element analyses for both natural and forced response computations, including displacement amplitudes and surface stresses. Results reveal that N-PRIME is capable of accurately predicting the dynamics of a blisk with severely large mistuning, along with small random mistuning throughout each sector. Also, N-PRIME can accurately capture modes with highly localized motions. A statistical analysis is performed to study the effects of random mistuning on both natural frequencies and forced responses and reveals that 1) new clusters of mistuned natural frequencies exist and cannot be treated as small deviations from nominal frequencies, and 2) unexpectedly high amplifications of forced responses exist, showing 3–5 times higher responses compared to the nominal system.

Reduced-Order Models for Blisks With Small and Large Mistuning and Friction Dampers

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.

Reduced Order Modeling of a Bladed Rotor with Geometric Mistuning: Alternative Bases and Extremely Large Mistuning

Reduced Order Modeling of a Bladed Rotor with Geometric Mistuning: Alternative Bases and Extremely Large Mistuning

Reduced-Order Modeling Approach for Blisks with Large Mass, Stiffness, and Geometric Mistuning

Deviations to the cyclic symmetry of turbomachinery blisks are common due to manufacturing, general wear, or foreign object damage. These deviations, called mistuning, can lead to dramatically increased forced responses of a few of the blades. The high computational cost associated with predictions for mistuned blisks has led to the development of various reduced-order modeling techniques. This paper introduces a new reduced-order modeling approach for both large and small mistuning. This approach makes use of a pristine-rogue-interface modal expansion (PRIME). The PRIME basis is a new basis composed of modes computed only from cyclic symmetry analyses.Bystrategicallyprojectinga finiteelementmodelfromthephysicalspaceontothenewbasis,itispossibleto accurately and efficiently model large mistuning. In addition, the new method can account for both large and small mistuning simultaneously, using an extension of the component mode mistuning approach. The methodology requires only sector-level calculations and therefore can be applied to highly refined, realistic models of industrial size.

Robust Strategies for Forced Response Reduction of Bladed Disks Based on Large Mistuning Concept

In this paper, robust maximum forced response reduction strategies based on a “large mistuning” concept are introduced, including both (i) random and (ii) deterministic approaches. An industrial bladed fan disk serves as an application example for a reliability assessment of the aforementioned strategies using two well-established tools for uncertainty analysis: (i) statistics and (ii) sensitivity and robustness. The feasibility and other practical aspects of implementing large mistuning as a means of preventing excessive forced response levels caused by random mistuning and ensuring the predictability of the response are discussed.

Local/Global Effects of Mistuning on the Forced Response of Bladed Disks

The focus of the present investigation is on the assessment and modeling of the local (spanning only a few blades) and global (encompassing the entire disk) effects of mistuning on the forced response of bladed disks. To this end, the concept of localization is first revisited and a new measure of this effect is introduced in terms of the number of blades the mistuning of which actually affects the forced response of a central blade. Using this new metric, it is demonstrated that high responding blades typically exhibit a high level of localization and that the reverse is not necessarily true. Thus, localization is not only disk dependent but also varies from blade-to-blade on the same disk. This observation is then used to validate a partial mistuning approach to the determination of the maximum amplitude of response over the entire population of disks. The results of this study indicate that the largest amplification due to the mistuning occurs at very strong blade-to-blade coupling levels, at the contrary of a general perception, but is associated with large mistuning levels. Finally, the above phenomenological observations are used to devise a modeling technique of both local and global components of mistuning. An example of application is presented that demonstrates the high accuracy of this approach through the entire blade-to-blade coupling domain.

Localization Phenomena in Mistuned Assemblies with Cyclic Symmetry Part I: Free Vibrations

An investigation of the effects of small structural irregularities on the dynamics of nearly periodic structures with cyclic symmetry is presented. The system studied may be regarded as a simple model of a continuously shrouded blade assembly accounting for one structural mode per blade. A key aspect of the approach is the use of perturbation methods that lead to a physical insight into the effects of mistuning. The study shows that the sensitivity to mistuning depends primarily upon the ratio of mistuning strength to coupling strength. For a small mistuning to coupling ratio, the mistuned system behaves like a perturbation of the corresponding tuned system, in which case mistuning has a relatively small effect on both the free and forced responses. On the other hand, for a large mistuning to coupling ratio (i.e., weak coupling), the mistuned system behaves like a perturbation of the corresponding decoupled mistuned system, in which case small mistuning dramatically changes the dynamics of the system. This paper, Part I, investigates the effects of small mistuning on the free response of the system. Specifically, it is shown that strong mode localization and eigenvalue loci veering phenomena occur in the weakly coupled system when mistuning is introduced. The effects of mistuning on the forced response are studied in the companion paper, Part II (Wei and Pierre, 1987).

Stability of Frequency Feedback Receivers Under Steps in Input Frequency

It is known that frequency feedback demodulators can show instability in their response to step changes (mistuning) in input frequency. This work reports on some mathematical analyses of this phenomenon as described by differential equations arising from simple IF and feedback filters in the demodulator. These equations are studied for local and global stability by geometric or phase-plane analysis, by means of Lyapunov functions, and by the topological Poincare-Bendixson methods. A typical result is for the case of no feedback filter and one-pole baseband analog of the IF filter, and states in physical terms that if the mistuning is not too big, specifically if |mistuning| < (half-power IF bandwidth)(1 + feedback gain) then solutions which are bounded away from zero amplitude approach the natural equilibrium point. Examples are given in which a sufficiently large mistuning makes the equilibrium point unstable.