Mistuning Analysis Research Articles

Detection of Cracks in Turbomachinery Blades by Online Monitoring

AbstractThe presence of a crack in a blade can change the natural frequencies of that blade. It has long been a goal to detect blade cracks by assessing the change in a measured vibration frequency of the blade over time. It has been found that prior frequency assessment methods can be less accurate than is desirable to reliably detect the relatively small frequency changes that are typically associated with blade crack sizes of practical interest. This paper describes a method in which potential temporal changes in the frequencies of individual blades are assessed by periodically analyzing complete rows of blades using mistuning analysis techniques that treat the blade rows as coupled systems, in contrast to other techniques that consider each blade individually in turn. This method, while computationally complicated and challenging, has been found to be capable of detecting blade root cracks that are much smaller than those that can be detected using other techniques. Moreover, this method has been demonstrated to detect cracks that are much smaller than the critical size for mechanical separation of the blade from the rotor. This improved frequency assessment technique has been used to identify more than 30 blades with frequency changes that were considered to be potential indicators of blade cracks. Subsequent inspections verified indications in all of those blades. In addition to providing operational guidance, the frequency change data were used to infer the time periods during which crack growth had occurred.

Numerical Investigation of a Mistuned Academic Bladed Disk Dynamics With Blade/Casing Contact

Abstract This contribution focuses on the combined analysis of mistuning and unilateral blade-tip/casing contacts. A two-dimensional (2D) phenomenological finite element (FE) model of an aircraft engine fan stage is considered. It is reduced by means of the Craig–Bampton component mode synthesis method and contact treatment relies on a Lagrange multiplier algorithm within an explicit time-integration scheme. Blade-tip/casing contacts are initiated through the deformed shape of a perfectly rigid casing. Mistuning is accounted for on the blades only. Monte Carlo simulations are carried out in both linear and nonlinear configurations, which allows to compare amplifications predicted in both context due to mistuning. Following a thorough convergence analysis of the proposed numerical strategy, the influence of mistuning level as well as the configuration of the external forcing are investigated. Presented results underline the detrimental consequences of mistuning in a nonlinear structural context, yielding even higher vibration amplifications than in a linear context. A cross-analysis between linear and nonlinear computations reveals that no correlation is found between linear and nonlinear amplifications which suggests that the effect of existing strategies to mitigate vibration amplifications within a linear context may not be suitable within a nonlinear context.

Forced Response Analysis of a Mistuned Blisk Using Noncyclic Reduced-Order Models

The importance of mistuning analysis lies on understanding the distribution of the vibrational energy around the blisk. The large vibration amplitudes of individual blades inherent in mistuned blisks reduces the high cycle fatigue margin significantly. It is therefore important to perform mistuning analyses at a high accuracy while keeping the computational cost at an acceptable level. Because numerous analyses with large amount of degrees of freedom models are commonly performed, it is frequent to employ reduced-order models such as to reduce the computational effort. In this paper, a unique way to address the reduced-order model is presented, where each blisk sector is attached as individual substructures with the free-interface approach known as Craig–Chang. This implementation is compared against a fixed-interface approach known as Craig–Bampton in terms of accuracy for disk- and blade-dominated modes. Neither of these approaches applies cyclic symmetry, making them more accurate in the presence of mistuning when the harmonic patterns are destroyed. Results show the high benefits of using the free-interface approach for a mistuned forced response analysis.

Novel Frame Model for Mistuning Analysis of Bladed Disk Systems

The work investigates the application of a novel frame model to reduce computational cost of the mistuning analysis of bladed disk systems. A full-scale finite element (FE) model of the bladed disk is considered as benchmark. The frame configuration for a single blade is identified through structural identification via an optimization process. The individual blades are then assembled by three-dimensional (3D) springs, whose parameters are determined by means of a calibration process. The dynamics of the novel beam frame assembly is also compared to those obtained from three state-of-the-art FE-based reduced order models (ROMs), namely: a lumped parameter approach, a Timoshenko beam assembly, and component mode synthesis (CMS)-based techniques with free and fixed interfaces. The development of these classical ROMs to represent the bladed disk is also addressed in detail. A methodology to perform the mistuning analysis is then proposed and implemented. A comparison of the modal properties and forced response dynamics between the aforementioned ROMs and the full-scale FE model is presented. The case study considered in this paper demonstrates that the beam frame assembly can predict the variations of the blade amplitude factors, and the results are in agreement with full-scale FE model. The CMS-based ROMs underestimate the maximum amplitude factor, while the results obtained from beam frame assembly are generally conservative. The beam frame assembly is four times more computationally efficient than the CMS fixed-interface approach. This study proves that the beam frame assembly can efficiently predict the mistuning behavior of bladed disks when low-order modes are of interest.

A Method to Reduce the Energy Localization in Mistuned Bladed Disks by Application-Specific Blade Pattern Arrangement

The focus of the paper is the analysis of mistuning, which are small deviations of the blade properties, e.g., due to manufacturing tolerances. The resonant amplitudes of turbine blades are very sensitive to these deviations which can lead to significantly increased vibratory response of some blades with the increased risk of high cycle fatigue. The main part of the paper discusses about the methods that can be used to find blade patterns which are insensitive to energy localization. The sensitivity of the forced response against harmonic mistuning, which is a harmonic alignment of the blades respective to the mistuning factors of the single blades, is examined. A previously developed reduced order model (ROM) is used to efficiently conduct Monte Carlo simulations (MCSs). Especially the influence of the variance of the harmonically mistuned blade patterns is discussed. On the basis of this analysis, rules are developed to suppress the energy localization. The rules are mainly focused on the alignment of the blades around the bladed disk. The approach also takes advantage of the special properties of harmonic mistuning patterns. An assignment of the blades to insensitive harmonic mistuning patterns with a specific variance and number of periods is used to reduce the maximum of the amplification factor of the forced response. A similar approach uses intentional mistuning patterns with different blade types which are aligned harmonically and are insensitive to an additional stochastic mistuning. In case of specific combinations of the dependent parameters, especially the variance of the mistuning factors and the number of periods, the energy localization can be reduced considerably.

大容量蒸気タービン静翼の振動特性と安定性に関する研究

A vane used in a low pressure end of steam turbine is usually fixed to an inner shroud and an outer ring by welding both ends. In such a vane structure, the damping in loading operation is comprised of the material damping and the aerodynamic damping, because the structural damping is very small. In such a steam turbine vane, the total damping may become negative, and the flutter may occur under the high loading condition. Therefore, in the design of the steam turbine vane, it becomes indispensable to evaluate the stability under the high loading condition. In this study, first, the vibration characteristics of steam turbine vane with grouped vane structure are examined in detail by use of the results of FE analysis. Second, the reduced order model (the equivalent spring-mass model) of the whole vane structure is assembled, based on the results of FE analysis. The stability analysis of the whole vane structure is carried out using the reduced order model, and the effect of the design parameter of the vane on the flutter is clarified. Finally, the mistuning analysis of the whole vane structure is carried out, using the Monte Carlo simulation, and the effect of the frequency deviation on the stability is clarified.

Forced Response Sensitivity of a Mistuned Rotor From an Embedded Compressor Stage

The frequency mistuning that occurs due to manufacturing variations and wear and tear of the blades has been shown to significantly affect the flutter and forced response behavior of a blade row. While tuned computational fluid dynamics (CFD) analyses are now an integral part of the design process, designers need a fast method to evaluate the localized high blade responses due to mistuning. In this research, steady and unsteady analyses are conducted on the second-stage rotor of an axial compressor, excited at the first torsion vibratory mode. A deterministic mistuning analysis is conducted using the numerical modal forces and the individual blade frequencies obtained experimentally by tip timing data. The mistuned blade responses are compared in the physical and traveling wave coordinates to the experimental data. The individual and combined impacts of frequency, aerodynamic, and forcing function perturbations on the predictions are assessed, highlighting the need to study mistuned systems probabilistically.

Probabilistic Vibration Analysis of a Mistuned Integrally Bladed Disk

Abstract. The objective of this paper is to probabilistically evaluate the effects of mistuned sectors on the dynamic characteristics of an integrally bladed disk (blisk). Small blade to blade physical variation in a disk is termed as mistuning. In this study, the dynamic characteristics of the perfectly tuned blisk were firstly analysed as a baseline. Secondly, a probabilistic approach is used for the mistuning analysis of a blisk. A reduced-order method named as the subset of nominal modes (SNM) was used to generate modes for a mistuned blisk from a cyclic perfectly tuned FE model without creating the full model. Furthermore, as only the modes with natural frequencies close to the modes of interest were considered, a relatively shorter computational time and a much smaller model size than a full blisk model is used. Therefore, the dynamic characteristics of the forced response for random mistuned blisks were obtained.

Accuracies of Reduced Order Models of a Bladed Rotor With Geometric Mistuning

The results from a reduced order model based on frequency mistuning are compared with those from recently developed modified modal domain analysis (MMDA). For the academic bladed rotor considered in this paper, the frequency mistuning analysis is unable to capture the effects of geometric mistuning, whereas MMDA provides accurate estimates of natural frequencies, mode shapes, and forced response.

Resonant Response of Mistuned Bladed Disks Including Aerodynamic Damping Effects

A mathematical model is developed to investigate the effects of aerodynamic damping on the maximum amplification factor of mistuned bladed disks. LINSUB, an inviscid linearized unsteady aerodynamic damping code, provides aerodynamic damping influence coefficients that are incorporated into a partial-mistuning model that takes advantage of mode localization. This mistuning analysis is then used to demonstrate the effects of aerodynamic damping on the maximum amplification factor of mistuned bladed disks. The relative importance of aerodynamic effects is determined by a comparison of aerodynamic damping and structural damping factors. It is shown that neglecting unsteady aerodynamics may result in the predicted optimal mistuning pattern not being optimum in the actual operating environment wherein unsteady aerodynamic effects are present.

The Effects of Aerodynamic Asymmetric Perturbations on Forced Response of Bladed Disks

Most of the existing mistuning research assumes that the aerodynamic forces on each of the blades are identical except for an interblade phase angle shift. In reality, blades also undergo asymmetric steady and unsteady aerodynamic forces due to manufacturing variations, blending, mis-staggered, or in-service wear or damage, which cause aerodynamically asymmetric systems. This paper presents the results of sensitivity studies on forced response due to aerodynamic asymmetry perturbations. The focus is only on the asymmetries due to blade motions. Hence, no asymmetric forcing functions are considered. Aerodynamic coupling due to blade motions in the equation of motion is represented using the single family of modes approach. The unsteady aerodynamic forces are computed using computational fluid dynamics (CFD) methods assuming aerodynamic symmetry. Then, the aerodynamic asymmetry is applied by perturbing the influence coefficient matrix in the physical coordinates such that the matrix is no longer circulant. Therefore, the resulting aerodynamic modal forces in the traveling wave coordinates become a full matrix. These aerodynamic perturbations influence both stiffness and damping while traditional frequency mistuning analysis only perturbs the stiffness. It was found that maximum blade amplitudes are significantly influenced by the perturbation of the imaginary part (damping) of unsteady aerodynamic modal forces. This is contrary to blade frequency mistuning where the stiffness perturbation dominates.

Reduced‐Order Model Development for Airfoil Forced Response

Two new reduced‐order models are developed to accurately and rapidly predict geometry deviation effects on airfoil forced response. Both models have significant application to improved mistuning analysis. The first developed model integrates a principal component analysis approach to reduce the number of defining geometric parameters, semianalytic eigensensitivity analysis, and first‐order Taylor series approximation to allow rapid as‐measured airfoil response analysis. A second developed model extends this approach and quantifies both random and bias errors between the reduced and full models. Adjusting for the bias significantly improves reduced‐order model accuracy. The error model is developed from a regression analysis of the relationship between airfoil geometry parameters and reduced‐order model error, leading to physics‐based error quantification. Both models are demonstrated on an advanced fan airfoil′s frequency, modal force, and forced response.

Analysis of Random Vibration for Mistuned Bladed Disk

When calculating the vibratory response of bladed turbine disks, it is assumed that all blades on a given disk are identical. This leads to the prediction that all blades experience the same amplitude. However, it has been shown experimentally that significant variations in these amplitudes occur for different blades on the same disk. These variations arise due to the effects of mistuning, which refers to small differences in blade characteristics. Although the mistuning effect on the resonant response has been studied systematically, few papers have researched the mistuning effect on the random vibration. This paper, first, presents the analysis method of the mistuning effect on the random vibration. Second, the mistuning analyses for the random vibration and the resonant vibration of the bladed turbine disk are carried out, and the both results are compared. From these results, the mistuning effect on the random vibration is clarified.

509 ミスチューンがある翼・ディスク系のランダム振動応答解析

When calculating the vibratory response of bladed turbine disks, it is assumed that all blades on a given disk are identical. This leads to the prediction that all blades experience the same amplitude. However, it has been shown experimentally that significant variations in these amplitudes occur for different blades on the same disk. These variations arise due to the effects of mistuning, which refers to small differences in blade characteristics. Although the mistuning effect on the resonant response has been studied systematically, few papers have researched the mistuning effect on the random vibration. This paper, first, presents the analysis method of the mistuning effect on the random vibration. Second, the mistuning analyses for the random vibration and the resonant vibration of the bladed turbine disk are carried out, and the both results are compared. From these results, the mistuning effect on the random vibration is clarified.

Blade Manufacturing Tolerances Definition for a Mistuned Industrial Bladed Disk

This paper deals with the characterization of the blade manufacturing geometric tolerances in order to get a given level of amplification in the forced response of a mistuned bladed-disk. The theory is based on the use of a nonparametric probabilistic model of blade random uncertainties. The dispersion parameters controlling the nonparametric model are estimated as a function of the geometric tolerances. The industrial application is devoted to the mistuning analysis of a 22 blades wide chord fan stage. Centrifugal stiffening due to rotational effects is also included. The results obtained validate the efficiency and the reliability of the method on three-dimensional bladed disks.

Symmetry Approach to Extension of Flutter Boundaries via Mistuning

A general framework is presented for analyzing and optimizing stability increases resulting from mistuning. The framework given is model independent and is based primarily on symmetry arguments. DifŽ cult practical issues are transformed to tractable mathematical questions. It is shown that mistuning analysis reduces to a block circular matrix eigenvalue/vector problem that can be solved efŽ ciently even for large problems. Similarly, the optimization becomes a standard linear constraint quadratic programming problem and can be solved numerically. Because the methods given are model-independent, they can be applied to various models and allow the researcher to easily conclude which models accurately capture mistuning and which do not. A simple quasisteady model for  utter in a cascade is used to illustrate and validate results in this paper.