Model Of Bladed Disk Research Articles

On a New Cyclic Symmetry Formulation Accounting for Boundaries Undergoing Nonlinear Forces

Abstract Although cyclic symmetry theory was initially developed for linear structures, the introduction of nonlinear forces on internal nodes of the fundamental sector does not affect the methodology. Nevertheless, the method is ill-suited when nonlinear forces are applied at the cyclic boundary. The purpose of this paper is to provide a complement to this theory and to propose a cyclic symmetry formulation for structures undergoing nonlinear forces at their cyclic boundary. A complete nonlinear cyclic formulation for such systems is derived in this work. The advantages of such an approach lies in the reduction of computational costs using the cyclic symmetry properties. The methodology is employed to characterize the dynamics of several mechanical systems. First, it is validated on simplified models of a cyclic system. Two nonlinearities are considered: a one-dimensional friction contact interface and a cubic nonlinearity. Both cases exhibit very different dynamics behaviors, yet, the results obtained with the new strategy are shown to be very accurate. Once the approach is validated, it is employed on a industrial finite element model of turbine bladed disk featuring contact interfaces between the blades' shrouds. The capability of the method to handle large systems is thus demonstrated. For all cases, periodic excitation are applied following either a traveling or standing wave shape for different engines orders.

Prediction of fatigue life of geometrically deviated steam turbine blades under thermo-mechanical conditions

This study explores the intricate factors affecting the fatigue life of steam turbine blades, encompassing steam flow-induced bending, centrifugal loading, vibration response, structural mistuning, and temperature-dependent influences. By focusing on the significance of mistuned steam turbine blades with varying blade geometries due to manufacturing tolerances, this research has paramount relevance for the power generation industries. By employing finite element analysis (FEA) software, a simplified, mistuned, scaled-down steam turbine bladed disk model was developed, considering temperature-dependent material properties. Initial FEA provided insights into the vibration characteristics and steady-state stress responses, with numerical stress distributions evaluated, which were subsequently exported to Fe-Safe software for fatigue life calculations based on centrifugal and harmonic sinusoidal pressure loadings. By investigating the vibration characteristics and response to geometric blade variations, this study affirmed the reliability of the developed FEA model, with findings highlighting the pronounced sensitivity of fatigue life to blade length, width, and thickness variations, in this order. However, in order to validate the developed numerical models, analytical life cycle assessments were calculated, which exhibited a discrepancy of under 3.37%, reinforcing the applicability of the developed numerical methodology to real-world scenarios involving mistuned steam turbine blades experiencing manufacturing deviations in blade geometry.

Sensitivity Analysis of Transient Forced Response of Mistuned Bladed Disks Under Complex Excitation and Varying Rotation Speed

Abstract During accelerations and decelerations of gas-turbine engines, the transient vibration effects in bladed disk vibration become significant and the transient response needs to be efficiently and accurately assessed. This paper develops an effective method for efficient calculations of the sensitivity of the transient vibration response for a mistuned-bladed disk under varying rotation speeds. The sensitivities are calculated concerning two major factors: (i) the blade mistuning and (ii) the rotor acceleration/deceleration rates. The method uses the large-scale finite element modeling of the bladed disk to accurately describe the dynamic properties of the mistuned bladed disk. The method can be applied for a case when the loading is the transient load, with the amplitude and frequency spectrum changing with rotation speed. The dependency of the modal characteristics on the rotation speed is included in the formulation of the sensitivity analysis method. Efficient reduced order models are used to reduce the computational cost, which can adapt to the effects of the varying rotation speed on the natural frequencies and mode shapes of the mistuned bladed disk under transient conditions. The expressions for the sensitivities are derived analytically which provides high accuracy and speed of the calculations. The method has been implemented in a computer code and thoroughly verified using a set of test cases. The studies of transient forced response sensitivity have been performed for realistic bladed disk models.

Damping and Stiffness Mistuning Effects in a Bladed Disk With Varied Disk Coupling

Abstract Component mode mistuning (CMM) is a well-known, well documented reduced order modeling technique that effectively models small variations in blade-to-blade stiffness for bladed disks. In practice, bladed disks always have variations, referred to as mistuning, and are a focus of a large amount of research. One element that is commonly ignored from small mistuning implementations is the variation within the blade-to-blade damping values. This work seeks to better understand the effects of damping mistuning by utilizing both structural and proportional damping formulations. This work builds from previous work that implemented structural damping mistuning reduced order models formulated based on CMM. A similar derivation was used to create reduced order models with a proportional damping formulation. The damping and stiffness mistuning values investigated in this study were derived using measured blade natural frequencies and damping ratios from high-speed rotating experiments on freestanding blades. The two separate damping formulations that are presented give very similar results, enabling the user to select their preferred method for a given application. A key parameter investigated in this work is the significance of blade-to-blade coupling. The blade-to-blade coupling study investigates how the level of coupling impacts damping mistuning effects versus applying average damping to the bladed disk model. Also, the interaction of stiffness and damping mistuning is studied. Monte Carlo simulations were carried out to determine amplification factors, or the ratio of mistuned blade responses to tuned blade responses, for various mistuning levels and patterns.

Numerical Investigation of Friction Induced Interaction of Flutter Modes in a Realistic Low Pressure Turbine Rotor

Abstract Flutter response of bladed disks is a quite an important problem in modern designs of low pressure turbines, but, even for tuned configurations, it is not fully understood yet. Some research work suggests that, in a tuned rotor, the flutter induced vibration state that sets in consists of just a single traveling wave mode, while others show flutter states composed of several traveling waves active at the same time. Many of these studies were performed using conceptual models, such as mass-spring systems or asymptotically reduced models. In this work, a high fidelity bladed disk model is integrated into the time domain for the case of a realistic aerodynamically unstable low pressure turbine. In this configuration, the only nonlinear effect is the friction present at the blade disk attachment, which is responsible for the saturation of the flutter growth, and it is also the only mechanism that allows the interaction among the different unstable traveling wave modes. A case with multiple unstable modes of the same modal family is studied first. After that, the study is extended to a case where modes of different modal families are unstable and have similar aerodynamic instability levels. The interaction of unstable modes from different modal families is studied, we believe, for the first time, as previous flutter analyses were restricted to the interaction of unstable modes from the same family.

Fatigue life prediction of mistuned steam turbine blades subjected to deviations in blade geometry

The blades of the steam turbine are subjected to bending of the steam flow, centrifugal loading, vibration response, and structural mistuning. These factors mentioned contribute significantly to the fatigue failure of steam turbine blades. Low pressure (LP) steam turbines experience premature blade and disk failures due to the stress concentrations in the root location of the blade of its bladed disk. This study of mistuned steam turbine blades subjected to variation in blade geometry will be of great significance to the electricity generation industry. A simplified, mistuned, scaled-down steam turbine bladed disk model was developed using ABAQUS finite element analysis (FEA) software. The acquisition of the vibration characteristics and steady-state stress response of the disk models was carried out through FEA. Such studies are very limited. Subsequently, numerical stress distributions were acquired and the model was subsequently exported to Fe-Safe software for fatigue life calculations based on centrifugal and harmonic sinusoidal pressure loading. Vibration characteristics and response of the variation of the geometric blade of the steam turbine were investigated. Natural FEA frequencies compared well with the published literature of real steam turbines, indicating the reliability of the developed FEA model. The study found that fatigue life is most sensitive to changes in blade length, followed by width and then thickness, in this order. Analytical life cycles and Fe-Safe software show a percentage difference of less than 4.86%. This concludes that the numerical methodology developed can be used for real-life mistuned steam turbine blades subjected to variations in blade geometry.

Prediction of fatigue life of mistuned steam turbine blades subjected to variations in blade geometry

There is a large number of power stations suffering from fatigue failures of the steam turbine blades. The steam turbine blades are also subjected to steam flow bending, centrifugal loading, vibration response, and structural mistuning. These mentioned factors significantly contribute to the fatigue failure of the steam turbine blades. Low-Pressure (LP) steam turbines experience premature blade and disk failures due to the stress concentrations at the blade root area of its bladed disk. Driven by the problems encountered by the steam power plant electricity generating utilities with regards to steam turbine blades fatigue failure, this study of the mistuned steam turbine blades subjected to variation in blade geometry will be of great significance to the electricity generation industry. A simplified, scaled-down mistuned steam turbine bladed disk model was developed using ABAQUS finite element analysis (FEA) software. Acquisition of the vibration characteristics and steady-state stress response of the disk models was performed through FEA. Thereafter, numerical stress distributions were acquired, and the model was subsequently exported to Fe-Safe software for fatigue life calculations based on centrifugal and harmonic sinusoidal pressure loading. The vibration characteristics and the response of the variation steam turbine geometric blade was conducted. The FEA natural frequencies compared well with published literature of the real steam turbines indicating reliability of the developed FEA model. The study found that the fatigue life is most sensitive to changes in blade length, followed by the width, and then the thickness, in this order. The analytical life cycles and Fe-Safe software shows the percentage difference of less than 4.86%. This concludes that the developed numerical methodology can be used for real-life mistuned steam turbine blades subjected to variations in blade geometry.

Mistuning identification and model updating approach for bladed disks based on blade tip timing

To solve the problem of online mistuning identification of bladed disks, this paper proposes a novel mistuning identification method based on proper orthogonal modes (POM-ID) and the response data. Firstly, the proposed method is analyzed by numerical simulation technology, and the influence of related parameters on the identification results is discussed. Compared with the traditional method, the effectiveness of the proposed method is verified. Secondly, the POM-ID method is applied to identify the mistuning parameters of an actual bladed disk based on the blade tip timing test data. Finally, the identification results of the rotating bladed disk are verified by using static test results, and the dynamic model of the bladed disk is updated. The accuracy of the updated model is confirmed by the aspects of natural frequency and vibration coupling effect. The results indicate that the natural frequency error of the modified model is less than 1%, and the vibration coupling law between blades is consistent with the test results. The mistuning identification method proposed in this paper is especially suitable for test data of the rotating bladed disk and is expected to be applicable for online dynamic updating of the bladed disk model.

Mechanism of interconnected synchronized switch damping for vibration control of blades

The Synchronized Switch Damping (SSD) is regarded as a promising alternative to mitigate the vibration of thin-walled structures in aero-engines, especially for blades or bladed disks. The common manner is to shunt the switch circuit independently to a single piezoelectric structure. This paper is aimed at exploring a novel way of using the SSD, i.e., the SSD is interconnected between two piezoelectric structures or substructures. The damping mechanism, performance, and effective range of the interconnected SSD are studied numerically and experimentally. First, based on a dual cantilever beam finite element model, the time domain and frequency domain modeling and solving methods of the interconnected SSD are deduced and validated. Then, the influence of the amplitude and phase relationship on the damping effect of the interconnected SSD is numerically studied and compared with the shunted SSD. A self-sensing SSD control board is developed, and experimental studies are carried out. The results show that the interconnected SSD establishes an additional energy channel between the corresponding piezoelectric structures. When the amplitudes of the two cantilever beams are different, the interconnected SSD balances the vibration level of each beam. When the amplitudes of the two cantilever beams are the same, if the appropriate interconnection manner is selected according to the phase, the resonance peak can be reduced by more than 30%. When the vibration is in-phase/out-of-phase, the damping generated by the interconnected SSD in a cross/parallel manner is even more significant than the shunted SSD. Furthermore, this novel connection scheme reduces the number of SSD circuits in half. Finally, for engineering applications, we implement the proposed damping technology to the finite element model of a typical dummy bladed disk. A piezoelectric damping ratio of 13.7% is achieved when the amount of piezo material is only 10% of blade mass. Compared with traditional friction dampers, the major advancements of the interconnected SSD are: (A) it can reduce the vibration level of blades without friction interface; (B) the space constraint is overcome, i.e., the vibration energy is not necessarily dissipated independently in one sector or through physically adjacent blades, and instead, the dissipation and transfer of vibrational energy can be realized between any blade pair. If a specific gating circuit is adopted to adjust the interconnection manner of the SSD, vibration mitigation under variable working conditions with different engine orders will be expected; (C) designers do not need to worry about the annoying nonlinearities related to working conditions anymore.

Fatigue life prediction of turbine blades with geometric imperfections made of stainless steel

This research addresses critical challenges faced by steam turbine blades, particularly in low-pressure (LP) turbines, where premature failures are common due to stress concentrations at the blade root area. The study introduces a numerical methodology aimed at predicting the life of mistuned steam turbine blades, with a focus on variations in blade geometry which have received limited exploration in existing literature. A simplified, scaled-down mistuned steam turbine bladed disc model was developed using Abaqus finite element software. Acquisition of steady-state stress response of the disc models was performed through finite element analysis (FEA). Thereafter, numerical stress distributions were obtained. Subsequently, within Companion software, a Monte Carlo simulation-based probabilistic approach was applied to evaluate and quantify uncertainties in fatigue life for 17 cases. This analysis considered an accepted manufacturing percentage scatter of ± 5 % for the steam turbine bladed disc. That was conducted by selecting mistuning (geometry variation) percentages as the random variables. The methodology demonstrated reliability, correlating well with literature-based and discrete fatigue life results. This study establishes the potential for accurately predicting the fatigue life of mistuned steam turbine blades using the developed methodology.

Numerical study of bladed structures with geometric and contact nonlinearities

This paper deals with the reduced order modeling of turbomachine bladed structures accounting for both geometric nonlinearities and nonlinear blade-tips/casing contacts in a numerically efficient way. A recently derived methodology, based on the concept of modal derivatives, is first used to study the contact interactions of a single rotating blade impacting a rigid casing. In this reduction procedure, nonlinear internal forces due to large displacements are evaluated using the stiffness evaluation procedure and contact is numerically handled with Lagrange multipliers. In-depth analyses, including clearance consumption computations and frequency analyses with a continuation procedure, are performed to understand and characterize the combined influence of contact and geometric nonlinearities on the blade’s dynamics. The methodology is then generalized to full bladed disks using Component Mode Synthesis techniques with fixed interfaces. Each sector of the cyclically symmetric structure is projected onto a reduction basis composed of Craig–Bampton modes and a selection of their modal derivatives. A second reduction allows reducing the cyclic boundary degrees-of-freedom. The methodology is first assessed using a harmonic excitation at blade-tips, without contact. When applied to the bladed disk subjected to contact interactions, the methodology allows identifying the main interaction speeds that can be detrimental for the engine integrity. Through this work, the numerical strategy is applied on an open industrial compressor bladed disk model based on the NASA rotor 37 in order to promote the reproducibility of results.

Novel Neural Network for Predicting the Vibration Response of Mistuned Bladed Disks

Inevitable mistuning in cyclic bladed disk structures would cause vibration amplification phenomena that seriously reduce the reliability of the bladed disk. The ability to accurately and quickly predict the dynamic responses is critical to investigating the dynamic behavior of the mistuned system. However, it is still challenging because the mistuned responses are extremely sensitive to the random mistuning parameters. In this work, a novel mistuned system deep neural network model (MS-DNN) is presented to predict the dynamic responses of mistuned bladed disks through the mistuning parameters for both the lumped parameter model and the large-scale finite element (FE) model, which decouples the vibration equations of the mistuned system and uses a neural network to replace the coupling process. MS-DNN is divided into two levels, namely, the blade and the disk. The blade-level neural networks are used for forward and backward propagation of mistuning parameters in the different blades, and the disk-level neural network is used to replace the physical coupling process in the disk of multiple mistuning parameters from individual blades, with data transmission between the neural networks via blade–disk boundary nodes. The expected physical response of the blade tip is predicted through MS-DNN. All neural networks in MS-DNN show high prediction accuracy on both training sets and unknown test sets. For the FE model of the industrial bladed disk, the effect of the number of boundary nodes selected as the data interface between neural networks on the prediction accuracy is also investigated. The results show that, for unknown test data, the predicted response has an value of 0.998 versus the actual response with an amplification factor error of less than 0.388%.

Nonlinear dynamic analysis of three-dimensional bladed-disks with frictional contact interfaces based on cyclic reduction strategies

In bladed disks, friction nonlinearities occurring within the contact between the blades and disk or between the blades and underplatform dampers are used to decrease the vibratory energy of the system and extend its lifespan. Modeling these nonlinearities and simulating their effects correctly are challenging: complex nonlinear phenomena such as stick, slip and separation may occur in the contact zones. Efficient numerical methods must be used to compute the dynamics of the system in a reasonable amount of time. This paper proposes a reduction strategy to tackle large cyclically symmetric finite-element models undergoing static preload from centrifugal effects and strong nonlinearities (friction and separation). It combines the nonlinear identification of the possible interacting nodal diameters with a linear component mode synthesis procedure. Its potential and performances are assessed through comparisons with some state-of-the-art methods. Complex and realistic nonlinear finite-element models of bladed disks with underplatform dampers and dovetail or fir-tree blade roots are used. The Dynamic Lagrangian Frequency Time algorithm is employed to capture the nonlinear effects. Using the proposed reduction method, the effect of underplatform dampers on bladed disk contact occurrence and damping efficiency is investigated.

Polynomial chaos expansion for permutation and cyclic permutation invariant systems: Application to mistuned bladed disks

This article deals with the stochastic modeling of industrial mistuned bladed disks. More specifically, a cost-efficient implementation of polynomial chaos expansion is proposed, it is dedicated to mathematical systems exhibiting permutation invariance or cyclic permutation invariance of their random variables. Significant gains are obtained in comparison to the classical implementation of polynomial chaos expansion since potentially costly evaluations of the investigated deterministic system are only required over a small subspace of the random space. The proposed methodology is detailed and validated on analytical test cases before it is applied to two mistuned bladed disks models. First, computations carried out with a simplified bladed disk model allow an in-depth comparison between Monte Carlo simulations, previously published results with a standard polynomial chaos expansion and the proposed methodology. The latter is then employed to assess the influence of mistuning on the eigenfrequencies and amplification magnification of an industrial compressor stage. It is evidenced that in comparison to the standard polynomial chaos expansion, the proposed methodology yields computational gains of the same order of magnitude as the ones obtained going from Monte Carlo simulations to polynomial chaos expansion. Alternately, the proposed methodology may be employed to significantly increase the accuracy of the standard polynomial chaos expansion while featuring an identical computational cost.

Analysis of Nonlinear Modal Damping Due to Friction at Blade Roots in Mistuned Bladed Disks

Abstract A method is proposed to analyze the modal damping in mistuned bladed-disk with root joints using large finite element models and the detailed description of frictional interactions at contact interfaces. The influence of mistuning on the dissipated energy for different blades on a bladed-disk and the modal damping factors for different vibration levels for any family of modes can be investigated. The dissipated energy and damping factors due to microslip are simulated by multitude of surface-to-surface elements modeling the friction contact interactions at root joints. The analysis is performed in the time domain, and an original reduction method is developed to obtain the results with acceptable computational times. The model reduction method allows the calculation of the modal damping of the mistuned assembly by evaluation of the energy dissipated at root joint of each individual blade using small parts of bladed disk sectors. The dependency of modal damping factor on blade mode shapes, engine-order excitation numbers, nodal diameter numbers, and vibration amplitudes is studied and the distributions of amplitude and dissipated energy on the mistuned bladed-disk are investigated using a realistic blade disk model.

The study on lashing ware location of vibration reduction model of detuned blade disc structure

The studies on vibration localization of periodic structure are mainly based on displacement modes at present, while the strain modes have not been applied to the research of vibration localization. Consequently, it was imagined to apply the strain modes to vibration localization based on a finite element model of bladed disk with shrouds, and the vibration localization factors based on strain modes and displacement modes were compared with the condition that the bladed disk is mistuning. The result shows that the vibration localization based on strain modes is much more sensitive to stiffness mistuning than the vibration localization based on displacement modes; what’s more, the strain mode shapes are different with the displacement modes at the same mode order. The result is important to the design of periodic structure with snubbing mechanism.

Uncertainty and global sensitivity analysis of bladed disk statics with material anisotropy and root geometry variations

Monocrystalline blades used in gas turbines exhibit material anisotropy, and the orientation of the anisotropy axis affects the deformation of the bladed disk. Considering the orientation of the anisotropy axis as a random design parameter, the uncertainty and sensitivity analysis for static blade deformations are presented for the first time. The following two cases are analyzed using a realistic bladed disk model with friction contacts: (i) deformation of the tuned bladed disks considering the uncertainty in anisotropy orientation and variations in fir tree root geometry and (ii) deformation of mistuned bladed disks with uncertain blade crystal orientations. For efficient uncertainty and sensitivity analysis, the applicability of the following surrogate models are explored: (i) an ensemble of regression trees (random forest) and (ii) gradient‐based polynomial chaos expansion. Faster convergence in statistical characteristics has been obtained using a surrogate model compared to that obtained from finite element model based Monte Carlo simulation. From sensitivity analysis, it has been inferred that the uncertainty in static displacements of a blade in a bladed disk is primarily due to the uncertainty in anisotropy angles of that blade itself and secondarily due to the interaction of different blade anisotropy angles.

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.

Vibration Characteristics of Rotating Mistuned Bladed Disks considering the Coriolis Force, Spin Softening, and Stress Stiffening Effects

Bladed disks of engine rotors usually operate at harsh conditions of high rotating speeds, which may lead to nonnegligible rotordynamic effects, including Coriolis force, spin softening, and stress stiffening effects. These effects on the vibration of mistuned bladed disks are seldom discussed in available investigations. In this paper, the vibration characteristics of rotating mistuned bladed disks are addressed by considering these rotordynamic effects. First, finite element (FE) models of bladed disks are used to obtain the governing equations of motion, and an efficient method for getting the stress stiffening matrix of sector model is developed. Then, the effective component‐mode mistuning method (CMM) is employed to create compact, yet accurate, reduced‐order models (ROMs). Finally, the models are validated and used to study the influences of Coriolis force, spin softening, and stress stiffening effects on the vibration of bladed disks with frequency mistuning factors. Numerical results show that these rotordynamic effects could significantly affect the vibrations of mistuning bladed disks, especially in the ranges of high speed, and should be carefully considered during analysis.

Rotational Speed-Dependent Contact Formulation for Nonlinear Blade Dynamics Prediction

Considering rotational speed-dependent stiffness for vibrational analysis of friction-damped bladed disk models has proven to lead to significant improvements in nonlinear frequency response curve computations. The accuracy of the result is driven by a suitable choice of reduction bases. Multimodel reduction combines various bases, which are valid for different parameter values. This composition reduces the solution error drastically. The resulting set of equations is typically solved by means of the harmonic balance method. Nonlinear forces are regularized by a Lagrangian approach embedded in an alternating frequency/time domain method providing the Fourier coefficients for the frequency domain solution. The aim of this paper is to expand the multimodel approach to address rotational speed-dependent contact situations. Various reduction bases derived from composing Craig–Bampton, Rubin–Martinez, and hybrid interface methods will be investigated with respect to their applicability to capture the changing contact situation correctly. The methods validity is examined based on small academic examples as well as large-scale industrial blade models. Coherent results show that the multimodel composition works successfully, even if multiple different reduction bases are used per sample point of variable rotational speed. This is an important issue in case that a contact situation for a specific value of the speed is uncertain forcing the algorithm to automatically choose a suitable basis. Additionally, the randomized singular value decomposition is applied to rapidly extract an appropriate multimodel basis. This approach improves the computational performance by orders of magnitude compared to the standard singular value decomposition, while preserving the ability to provide a best rank approximation.