Geometric Mistuning Research Articles

Geometric mistuning identification and finite element model updating methods for bladed disks

Dynamic analysis results of a blisk are sensitive to inevitable geometrical mistuning, and the high-fidelity finite element model (FEM) of real blisk is of paramount importance to accurately predict structural dynamics. However, it is still challenging to accurately identify geometric mistuning and obtain a high-fidelity FEM due to the complexity and stochastic nature of geometric mistuning. In this paper, considering various types of geometrical mistuning, both a novel geometric mistuning identification and automatic finite element model updating (FEMU) method for real blisk using measured point cloud data are presented to reduce the discrepancies between the real blisk and the numerical model. The geometric surface of the real blisk is digitized to the point cloud data by a non-contact optical measurement technique. Three implementation steps of the method are introduced: 1) morphological feature detection from measured point cloud data of blisk; 2) surface nodes space mapping of FEM using isoparametric transformation; and 3) mesh shape optimization for FEMU. Eventually, two experimental cases are investigated: a simplified blisk and an industry blisk. The robustness of pre-processing algorithms for geometrical feature detection with measurement noise is also conducted. Results show that the blades with small mistuning, large deformations, and damages in the tip and leading edges are accurately identified and updated using the proposed method. Compared with measured natural frequencies for updated FEM, the maximum error is less than 3.38‰.

Sector Mode Assembling Reduction Technique for High-Fidelity Blisk Models with Geometry Mistuning

Blade mistuning in blisks arises primarily from the scatters of blade geometry profiles caused by manufacturing tolerance, in-service wear, blade repairs, etc. There is a recent trend to capture the blade-to-blade geometry variances through precise geometry measurements by a 3D optical scanning system in order to obtain an improved blade geometric mistuning evaluation capability. However, this usually leads to prohibitive computational costs due to the large-scale, high-fidelity industrial blisk finite element models. This paper develops an original model reduction approach, Sector Mode Assembling Reduction Technique (SMART), specifically for the high-fidelity blisk model fully mistuned by blade geometric variances, with either topologically compatible or incompatible blade meshes. The basic idea of SMART is to construct the sector-level reduction mode basis by strategically assembling the truncated cyclic modes independently computed for each “isolated” sector with assumed cyclic symmetry at the sector interfaces. Benefiting from the block structure of the SMART mode basis, the reduced-order models are derived by a series of sector-level projections with a relatively low memory requirement and computational cost. Another hidden benefit is that the SMART approach enables efficient structural modification predictions of the global blisk modes because only the modes of the sectors undergoing blade modification need to be re-evaluated and replaced in the SMART mode basis. The SMART approach is applied into a high-fidelity “as-measured model” of a blended blisk, constructed upon the geometry measurement by the state-of-the-art 3D optical scanning technology. It is fully demonstrated that the reduced-order model derived by SMART, featured by a minimal size, is able to reproduce the dynamics of the full-order as-measured blisk model with high accuracy.

Numerical Methods for Calculating Component Modes for Geometric Mistuning Reduced-Order Models

AbstractGeometric mistuning models formulated from component mode synthesis methods often require the calculation of component modes, particularly constraint and fixed interface normal modes (FINMs), during substructuring. For integrally bladed rotors, these calculations are required for each sector. This paper proposes methods that reuse information garnered from solving the constraint modes of a single sector on the remaining sectors to reduce memory requirements and solution times. A mesh metamorphosis tool is used to ensure finite element models match geometry obtained from a three-dimensional optical scanner. This tool also produces a common mesh pattern from sector-to-sector. This is exploited to produce common permutation matrices and symbolic factorizations of sector stiffness matrices that are proposed for reuse in solving subsequent constraint modes. Furthermore, a drop tolerance is introduced to remove small values during constraint mode calculation to reduce memory requirements. It is proposed to reuse this dropping pattern produced from a single sector on the remaining sectors. Approaches are then extended to a parallel processing scheme to propose effective matrix partitioning methods. Finally, information gathered during the constraint mode calculations is reused during the solution of the FINMs to improve solution time. Results show reusing permutation matrices and symbolic factorizations from sector-to-sector improves solution time and introduces no error. Using a drop tolerance is shown to reduce storage requirements of a constraint mode matrix, while reusing the dropping pattern introduces minimal error. Similarly, reusing constraint mode information in calculating normal modes greatly improves the performance.

Integrally Bladed Rotor Mistuning Identification and Model Updating Using Geometric Mistuning Models

Abstract Nonuniform manufacturing variations and uneven usage wear and damage, referred to as mistuning, can drastically alter the dynamic response of integrally blade rotors (IBRs). Optical scanners, combined with finite element model (FEM) mesh metamorphosis algorithms, have provided capabilities to create analytical models that reduce the effect of geometrical uncertainties in numerical predictions. However, deviations in material properties cannot be obtained via optical scanning, so additional approaches are needed. A geometric mistuning reduced-order model (ROM) is developed and modified to solve for unknown IBR sector eigenvalues that are linearly proportional to elastic modulus. The developed approach accounts for both proportional and nonproportional mistuning and allows updating of the elastic modulus for each sector in the ROM. Different tuned and mistuned modal reduction procedures are employed to understand the implications of each for identifying mistuning. Simulated test data with known inputs indicate the efficiency and accuracy of the method and improvements over using a traditional, tuned mode approach. The developed methods are then extended to bench-level traveling wave excitation (TWE) data to discern how sector frequencies vary due to geometry and modulus mistuning.

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.

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.

Application of Tuned Vibration Absorber Concept to Blisk Ring Dampers: A Nonlinear Study

AbstractIn this study, a novel design for ring dampers is proposed, where the concept of tuned vibration absorbers is leveraged to substantially increase damper effectiveness while minimizing potential stresses near the blade root. Tuned absorbers have been used in the past to reduce the forced response amplitudes of both mechanical and civil structures. The absorber natural frequency is tuned to the targeted frequency of the host structure where it is attached. The vibration reduction mechanism relies on energy transfer from the host structure to the absorber. The novel design technique proposed here uses a vibration absorber approach to achieve energy transfer from the blisk to the damper, which leads to larger damper motion. This enables energy dissipation due to friction, reducing vibrations even in blade-dominated modes. An academic finite element model of a blisk with a ring damper is used to demonstrate the novel tuned damper concept and design technique. The geometric mistuning of the damper due to the presence of a gap in the ring structure is also taken into account. The results demonstrate the validity of the proposed tuned damper concept, showing a substantial vibration amplitude reduction compared to the linear baseline results, in which the damper is not tuned or absent.

Geometric Mistuning Reduced-Order Model Development Utilizing Bayesian Surrogate Models for Component Mode Calculations

AbstractIntegrally 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.

Parametric Reduced Order Models for Bladed Disks With Mistuning and Varying Operational Speed

A considerable amount of research has been conducted to develop reduced order models (ROMs) of bladed disks that can be constructed using single sector calculations when there is mistuning present. A variety of methods have been developed to efficiently handle different types of mistuning ranging from small frequency mistuning, which can be modeled using a variety of methods including component mode mistuning (CMM), to large geometric mistuning, which can be modeled using multiple techniques including pristine rogue interface modal expansion (PRIME). Research has also been conducted on developing ROMs that can accommodate the variation of specific parameters in the reduced space; these models are referred to as parametric reduced order models (PROMs). This work introduces a PROM for bladed disks that allows for the variation of rotational speed in the reduced space. These PROMs are created by extracting information from sector models at three rotational speeds, and then the appropriate ROM is efficiently constructed in the reduced space at any other desired speed. This work integrates these new PROMs for bladed disks with two existing mistuning methods, CMM and PRIME, to illustrate how the method can be readily applied for a variety of mistuning methods. Frequencies and forced response calculations using these new PROMs are compared to the full order finite element calculations to demonstrate the effectiveness of the method.

Mistuning Identification Approach With Focus on High-Speed Centrifugal Compressors

Blade vibrations are one of the main cost drivers in turbomachinery. Computational blade vibration analysis facilitates an enormous potential to increase the productivity in the design of bladed components. Increasing computing power as well as improved modeling and simulation methods leads to comprehensive calculation results. This allows for a more precise prediction and assessment of experimental data. Usually, in the field of turbomachinery, identical blades are assumed to lower the required computational resources. However, mistuning is unavoidable, since small deviations due to the manufacturing process will lead to slightly different blade behavior. Potential effects such as mode localization and amplification can be treated statistically and have been thoroughly studied in the past. Since then, several reduced order models (ROMs) have been invented in order to calculate the maximum vibration amplitude of a fleet of mistuned blisks. Most commonly, mistuning is thereby modeled by small material deviations from blade to blade, e.g., Young's modulus or density. Nowadays, it is common knowledge that the level of manufacturing imperfection (referred as level of mistuning) significantly influences mode localization as well as vibration amplification effects. Optical measurements of the geometric deviations of manufactured blades and converting to a high-fidelity finite element (FE) model make huge progress. However, to the knowledge of the authors, there is no reliable method that derives a characteristic quantity from the geometric mistuning, that fits into the mentioned statistically approaches. Therefore, experimental data are needed to quantify the level of mistuning. Several approaches, which isolate blade individual parameters, are used to identify the dynamic behavior of axial compressors and turbines. These methods can be applied to medium-speed centrifugal turbine wheels but tend to fail to evaluate high-speed compressor with splitter blades. This paper briefly presents the original approach and discusses the reasons for failure. Thereafter, a new approach is proposed. Finally, the level of mistuning and important quantities to perform a statistical evaluation of a high-speed compressor is shown.

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.

Comparison of parametric and non‐parametric methods for bladed disks with random geometric mistuning

AbstractThis paper presents different strategies to quantify the influence of random geometric mistuning on the dynamic behavior of bladed disk. These strategies are compared for different correlation lengths of the random field describing the blade geometry. The results allow to determine the kind of random geometric mistuning for which these strategies are appropriate. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Estimation of Forcing Functions on a Mistuned Bladed Rotor From Harmonic Response

This paper deals with the estimation of forcing functions on a mistuned bladed rotor from measurements of harmonic response via Kalman filter (KF) in time domain. An unique feature of this approach is that the number of estimated variables can be far greater than the number of measurements. The robustness of this method to measurement errors is shown. It is also shown that direct prediction of amplitude and phase of sinusoidal force vector from input/output frequency response function has a large amount of errors in the presence of unavoidable measurement noise. Numerical examples contain both frequency mistuning and geometric mistuning.

Reduced-Order Models of Blisks With Small Geometric Mistuning

A technique for generating reduced-order models (ROMs) of bladed disks with small geometric mistuning is proposed. Discrepancies in structural properties (mistuning) from blade to blade can cause a significant increase in the maximum vibratory stress. The effects of mistuning have been studied over the past few decades. Many researchers have studied the dynamic behavior of mistuned bladed disks by using ROMs. Many of these techniques rely on the fact that the modes of a mistuned system can be approximated by a linear combination of modes of the corresponding tuned system. In addition, the tuned system modes have been modeled in component mode mistuning by using modal participation factors of cantilevered blade modes. Such techniques assume that mistuning can be well modeled as variations in blade-alone frequencies. However, since geometric deformations contain stiffness and mass variations, mistuning can no longer be captured by cantilevered blade modes alone. To address this, several studies have focused on large and small geometric mistuning. These studies exploited the difference between tuned (with perturbed geometry) and nominal tuned mode shapes. In this work, we extend on that approach and devote particular attention to the development of ROMs of bladed disks with small geometric mistuning. The methodology requires only sector-level calculations and therefore can be applied to highly refined, realistic models of industrial size.

Geometric uncertainties in finite element models of mistuned bladed disks

AbstractThis paper presents a method to develop reduced‐order models of bladed disks with geometric mistuning, which is based on polynomial choas approximation of modes of random eigenvalue subproblems. This method allows to reduce mass and stiffness matrices without evaluating the deviation in these matrices due to mistuning. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Estimation of Forcing Function for a Geometrically Mistuned Bladed Rotor Via Modified Modal Domain Analysis

Modified modal domain analysis (MMDA) is a method to generate an accurate reduced-order model (ROM) of a bladed disk with geometric mistuning. An algorithm based on the MMDA ROM and a state observer is developed to estimate forcing functions for synchronous (including integer multiples) conditions from the dynamic responses obtained at few nodal locations of blades. The method is tested on a simple spring-mass model, finite element model (FEM) of a geometrically mistuned academic rotor, and FEM of a bladed rotor of an industrial-scale transonic research compressor. The accuracy of the forcing function estimation algorithm is examined by varying the order of ROM and the number of vibration output signals.

Identification of mistuning and model updating of an academic blisk based on geometry and vibration measurements

In this work, an experimental modal analysis is performed on an academic bladed disk using a base excitation to identify the mistuning of each blade. Optical measurement is used to obtain the exact geometry of the structure and to be able to associate geometric mistuning to each blade. Differences are observed between the experimentally identified mistuning and the geometric mistuning. Since the bladed disk is a one-piece structure, there are no welded connections between the blades and the disk and the material properties can be assumed to be uniform. It can be shown that these differences come from non uniform clamping conditions, and that this mistuning is of the same order of magnitude than the variations in the geometry of the structure. It follows that the precise characterization of mistuning for industrial structures is in practice illusory because of the numerous factors introducing mistuning, such as the clamping conditions, aerodynamic damping, wear in service, etc.

Mistuned Response Prediction of Dual Flow-Path Integrally Bladed Rotors With Geometric Mistuning

The geometric mistuning problem is investigated for dual flow-path integrally bladed rotors (DFIBRs) by outlining two methods that explicitly account for blade geometry surface deviations. The methods result in reduced-order models (ROMs) that are a reduced form of a parent Craig–Bampton component mode synthesis (CB-CMS) model. This is accomplished by performing a secondary modal analysis on different degrees of freedom (DOF) of the parent model. The DFIBR is formulated in cyclic symmetry coordinates with a tuned disk and ring and blades with small geometric deviations. The first method performs an eigen-analysis on the constraint DOF that provides a truncated set of interface modes, while the second method includes the disk and ring fixed interface normal mode in the eigen-analysis to yield a truncated set of ancillary modes. Utilization of tuned modes have the benefit of being solved in cyclic symmetry coordinates and only need to be calculated once, which offers significant computational savings for subsequent mistuning studies. Each geometric mistuning method relies upon the use of geometrically mistuned blade modes in the component mode framework to provide an accurate ROM. Forced response results are compared to both the full finite element model (FEM) solutions and a traditional frequency-based approach outlined in a previous effort. It is shown that the models provide highly accurate results with a significant reduction in solution time compared to the full FEM and parent ROM.

Modified Modal Domain Analysis of a Bladed Rotor Using Coordinate Measurement Machine Data on Geometric Mistuning

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.