Bladed Disk Vibration Research Articles

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.

Accurate Interpolation of the Dependency of Modal Properties on the Rotation Speed for the Transient Response Analysis of Bladed Disks

Abstract During fast gas-turbine engine acceleration and deceleration, the transient vibration effects in bladed disk vibration become significant and the transient response needs to be calculated. In this paper, an effective method is developed for efficient calculations of the transient vibration response for mistuned bladed disks under varying rotation speeds. The method uses the large-scale finite element modeling of the bladed disks allowing the accurate description of the dynamic properties of the mistuned bladed disks. The effects of the varying rotation speed on the natural frequencies and mode shapes of a mistuned bladed disk and its effects on the amplitude and the spectral composition of the loading are considered. The dependency of the modal characteristics on the rotation speed is based on the evaluation of these characteristics at reference points followed by the interpolation to obtain values at any rotation speed from the operating range. A new method has been developed for the interpolation of mode shapes while preserving the orthogonality and mass normalization of the mode shapes. The method of mode shape interpolation is elaborated for tuned and mistuned bladed disks. The accuracy and efficiency of the method are demonstrated on test examples and on analysis of transient forced response of realistic bladed disks.

Application of Improved Dynamic Substructure Finite Element Model-Based State-Space Techniques in Mistuned Blisks

Aeroengine is a complex mechanical equipment, and it works at high temperature, pressure, rotational-speed, and severe loads. One of the core problems is that the vibration and mistuning of bladed disk lead to failure and affect the safety and reliability of aeroengine. Previously, one sector taken as the research object is not suitable; the integrally mistuned bladed disk (blisk) is taken as the research object is very necessary; however, the computational efficiency of mistuned blisk is very low. Therefore, a reduced-order model approach, i.e., an improved dynamic substructure finite element model-based state-space technique (IDSFEM-SST), is proposed to investigate the mistuned blisk. Firstly, the reduced-order substructure finite element model is established by this method, and then, the modal frequencies and modal strain energy amplitudes are investigated. Secondly, the maximum displacement responses are analyzed. Finally, the computational efficiency and accuracy of mistuned blisk via IDSFEM-SST is compared with that of the classical dynamic substructure finite element model and the high-fidelity finite element model to verify the effectiveness of this approach. This study has significance to the dynamic research and engineering practices for complex mechanical structures.

Dual-Connected Synchronized Switch Damping for Vibration Control of Bladed Disks in Aero-Engines

An enhanced SSDI (synchronized switch damping on inductor) approach is proposed to suppress the vibration of bladed disks in aero-engines. Different from the authors’ former work (MSSP, 2017; JIMSS, 2018) where a local SSDI circuit is shunted to the piezoelectric materials at each blade sector, in this work two blade sectors are interconnected by a shared SSDI circuit. In this way, the switching action of SSDI is triggered by the relative displacement between two blade sectors. The feasibility of the dual-connected SSDI is numerically examined by a 2-DOF (degree-of-freedom) mechanical system, and further experimentally validated on a single-beam and a double-beam system. Results show that the damping performance increases with the amplitude of relative displacement. This feature is especially favorable for the application of blisks where the blade normally vibrates in different amplitudes and phases. Eventually, we conduct numerical simulation on the forced response of mistuned bladed disk undergoing travelling wave excitation. Results show that the dual-connected configuration can reduce at least half the number of switching shunts while maintain nearly the same performance as the conventional (local) SSDI.

Multi-Wave Vibration Caused by Flutter Instability and Nonlinear Tip-Shroud Friction

Abstract Measurements revealed the contribution of multiple traveling waves to the flutter vibrations of bladed disks. Saturated flutter vibration, whether in this multi-wave or in its better-understood single-wave form, is a nonlinear phenomenon. However, it is still not understood of what physical origin the relevant nonlinearities are, and under what conditions single-wave or multi-wave flutter vibration occurs. Recent theoretical work suggests that multi-wave flutter vibration can be explained by strongly nonlinear frictional interblade coupling. The verity of this hypothesis is strictly limited by the simplicity of the considered model, namely, a cyclic chain of seven oscillators with frictional coupling and rather unrealistic aeroelastic behavior. In this work, it is demonstrated that nonlinear dynamical contact interactions at tip-shrouds are a likely cause for the observed multi-wave flutter vibration. To this end, a more realistic structural turbine blade row model with a more realistic aeroelastic behavior is considered. Some insight into its intriguing dynamics, dependence of limit states on initial conditions, and eigenvalue placement is provided. For instance, it is shown that there is an intimate relation between internal combination resonance conditions of certain traveling wave modes and the spectral content of single- and multi-wave flutter oscillations.

Eigenvalue and eigenvector derivatives of fractional vibration systems

Dynamic characterizations of fractional vibration systems have recently attracted significant research interest. Increasingly, successful applications of fractional derivatives have been found to the modeling of mechanical damping, vibration transmissions, improved fractional vibration controls and nonlinear vibration analyses. To facilitate further development, the eigenvalue problem including its derivatives, which are the central issues of vibration analysis, have to be fully established. This paper examines how eigenvalue and eigenvector derivatives of fractional systems can be derived when system matrices become functions of physical design parameters. First, new important orthonormal constraints are proposed since the modes are no longer orthonormal to the mass matrix, in this case due to its complex and frequency dependent nature. Next, new methods of eigenvector derivatives are developed for distinct eigenvalues for the cases of complete, incomplete and single mode modal data. Realistic and practical FE models incorporating fractional derivatives in the form of viscoelastic supports are employed to demonstrate the numerical accuracy and computational efficiency of the proposed methods. However, when repeated eigenvalues are considered due to structural spatial symmetries, the eigenvector space degenerates and further differentiation of system matrices are required in order to uniquely determine the eigenvector derivatives. Consequently, a new and effective general method is developed which can be applied to compute eigenvector derivatives of repeated eigenvalues with any multiplicity m. A simplified turbine bladed disk vibration model which is known to have repeated eigenvalues due to its cyclic symmetry, is then used to demonstrate the accuracy and salient features of the proposed 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.

Vibration Absorbers for a Mistuned Bladed Disk

Mistuning refers to variations in modal properties of blades due to manufacturing tolerances and material defects. This can result in the amplification of a blade vibratory amplitude. This paper deals with the design of vibration absorbers for a mistuned bladed disk. First, the basic theory is established for undamped vibration absorbers using a single-mode model for each blade. Then, it is extended to include a multiple mode model of each blade and disk dynamics. The impact of mistuning on the bladed disk vibration is examined in the presence of undamped absorbers via Monte Carlo simulations. It is found that vibration absorbers can be an effective method to counter the detrimental effects of mistuning.

Investigation of mistuning impact on vibration of rotor bladed disks

Mistuning often reduces the fatigue life of bladed disks. The objective of this study is to determine the degree of influence of various types of mistuning on bladed disk vibration. It is also important to determine how the position of the detuned blades in the bladed disk affects the vibrations. The results of experimental and numerical analysis of mistuned bladed disk vibration are presented. The authors investigated the effect of various types of mistuning (geometry, mass, etc.) on the free vibrations of the bladed disk. The worst cases with minimum mistuning and maximum localization were identified. The developed algorithms for calculating of mistuned bladed disks vibration and obtained results can be used, when designing turbomachines rotors.

Types of aeroelastic vibrations of turbomachinery bladed discs

A real process was analyzed for presence of self-oscillations. Absence of self-oscillations was revealed as its attributes such as progressive wave and beating were not discovered. A model of oscillatory process was proposed on the basis of the kind of oscillograms and the linear superposition theory for several simultaneously excited oscillation modes. In accordance with the theory two vibrations with frequencies equal to 919 and 469Hz were summed up. As a result of addition of these frequencies and comparison of the derived oscillograms with the real process it was found that the proposed model agrees with the original well enough. Moving further into the process, a third frequency component equal to 1002Hz originates in addition to those of 465 and 918 Hz. Addition of three frequencies carried out similarly to the first example also yields a satisfactory result. Oscillograms of real and model processes have a similar appearance. The frequency components calculated on the basis of the oscillogram of the model process practically coincided with the real process frequencies. Classification of aeroelastic oscillations was analyzed in order to determine the oscillatory type of the process under investigation. Three main phenomena were distinguished: forced oscillations, flutter and non-synchronous oscillations. Almost all foreign engine companies faced the latter phenomenon, sometimes mistaken for flutter. The analyzed example was labeled as non-synchronous oscillations according to features of non-synchronous oscillations, namely, aliquant rotation frequency and resonant behavior of oscillations.

On the Interaction of Multiple Traveling Wave Modes in the Flutter Vibrations of Friction-Damped Tuned Bladed Disks

It is well-known that flutter vibrations of bladed disks can be saturated by dry friction. Previous theoretical investigations indicated that the steady-state, friction-damped flutter vibrations of tuned bladed disks are always dominated by a single traveling wave component, even if multiple traveling wave forms are unstable. This contrasts recent experimental investigations where multiple traveling wave forms were found to participate at steady state. In this paper, we demonstrate that this phenomenon can be explained by nonlinear frictional interblade coupling. To this end, we consider a simple phenomenological model of a bladed disk with frictional intersector coupling and two unstable traveling waves forms. Vibrations occur not only in the form of limit cycle oscillations (periodic) but also in the form of limit torus oscillations (quasi-periodic). It is shown how the limit state depends on the initial conditions, and that the occurrence of multiwave flutter depends on the proximity of the complex eigenvalues of the associated unstable waves. Finally, by computing the limit torus oscillation with a frequency-domain method, we lay the cornerstone for the systematic prediction of friction-saturated flutter vibrations of state-of-the-art bladed disk models.

Influence of vibration conectedness on eigenmode shapes of bladed disks with non-identical blades

The paper discusses the vibrations of bladed disks with non-identical blades. Eigenvalues of this disks due to violation of rotational symmetry is always the case of the harmonic distortion of the distribution of the displacement along the circumference (the eigenvalues distortion). It has been established that the magnitude of distortion, and hence the degree of disruption of rotational symmetry of a wheel depends not only on how much different vanes from one another, but also from connection oscillations. By the calculation and experimental way it is studied distortion of eigenvaluesof bladed disks in violation of rotational symmetry with decreasing vibration connectedness up (by increasing the thickness of the disk) to its complete disappearance on an example of a model bladed disk.Introduced a coefficient of quantitatively assessing the value of the degree of violation of rotational symmetry. As an example of mode shapes represented by the bladed disks of the NK-12 compressor. A decrease in the vibration connectedness leads to a distortion in the beginning of the distribution of the harmonic offsets, and then to localize forms. It was noted that localized forms are not self-equilibrated and may lead to excitation of oscillations of the rotor.

Nonsynchronous vibrations of turbomachinery bladed discs

The paper brings forward the method of determining nonsynchronous vibrations of turbomachinery bladeddiscs, based on the hypothesis of mechanical generator. Turbomachinery stages represented as a mechanicalgenerator open up new opportunities for test results interpretation for the purpose of gas-turbine engine mountsdevelopment and make it possible to improve the reliability of diagnostics of nonsynchronous vibrations andtheir occurrence and development mechanism. Turbomachinery bladed discs represented as a mechanical generatorcomponent part revealed that nonsynchronous vibrations of turbomachinery bladed disc are determined bydesign features of bladed disc and vane. The paper contains some examples of spectra for fan strain-gaugingresults, on the basis of which the analysis of nonsynchronous vibrations of turbomachinery bladed discs wascarried out. Vibrations of bladed disc with forward and backward strain waves having coefficients k1 and k2, respectively,are considered as a mode of bladed disc being a part of the mechanical generator, which vibrationmode is presented by the algebraic sum of broken coefficients. The algebraic sum of these coefficients equalswith a ratio of vibration frequency of bladed disc as a mechanical generator part to rotation frequency. The introducedgenerator of mechanical vibrations model can be applied to the analysis of nonsynchronous vibrationswhen processing results of turbomachinery bladed discs rig tests.

Asymptotic description of damping mistuning effects on the forced response of turbomachinery bladed disks

The effect of mistuning on the vibration of bladed disks has been extensively studied in the past 30 years. Most of these analyses typically cover the case of small variations of the elastic characteristics (mass and stiffness) of the blades. In this work we study the not so frequent case of the forced response of a bladed-disk with damping mistuning. A reduced order model, that provides a simplified description of the dynamics of the system, is derived from the full problem using a consistent asymptotic procedure. The resulting simplified model describes quantitatively the effect of blade to blade damping variation on the vibrational response of the system, and gives precise information on the underlying mechanisms involved in the action of damping mistuning. The reduced order model is applied to compute the forced response of an integrated bladed-disk with mistuning damping, and to locate the damping mistuning pattern that gives maximum amplification of the response. A detailed finite element model is also used to compute the forced response of the mistuned blisk, and the results show an excellent agreement with the predictions obtained from the reduced order model.

A Study of Nonlinear Vibrations in a Frictionally Damped Turbine Bladed Disk With Comprehensive Modeling of Aerodynamic Effects

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

An adaptive control strategy based on passive piezoelectric shunt techniques applied to mistuned bladed disks

This paper mainly presents an adaptive control strategy based on passive piezoelectric shunt techniques for mistuned bladed disks. Resonant shunted piezoelectric patches are attached onto the disk between adjacent blades to reduce the blade vibration through blade–disk coupling. This piezoelectric network is of engineering interest since piezoelectric transducers are placed out of the main stream in turbomachinery. Piezoelectric damping and piezoelectric mistuning are exploited to minimize blade mistuning effects in this study. Initially an optimal piezoelectric mistuning obtained by genetic algorithm optimization is introduced into a given blade-mistuned disk to achieve a further blade vibration mitigation. In reality the blade mistuning pattern is not constant in the long run. Benefiting from the manageability and controllability of piezoelectric shunt circuits, this adaptive control strategy is to adjust the “optimal” piezoelectric mistuning pattern according to the slowly varying blade mistuning pattern. Numerical simulation reveals that a good performance is achieved with respect to reducing the vibration of time-variant mistuned bladed disks.

Dual Time Stepping Algorithms With the High Order Harmonic Balance Method for Contact Interfaces With Fretting-Wear

Contact interfaces with dry friction are frequently used in turbomachinery. Dry friction damping produced by the sliding surfaces of these interfaces reduces the amplitude of bladed-disk vibration. The relative displacements at these interfaces lead to fretting-wear which reduces the average life expectancy of the structure. Frequency response functions are calculated numerically by using the multi-harmonic balance method (mHBM). The dynamic Lagrangian frequency-time method is used to calculate contact forces in the frequency domain. A new strategy for solving nonlinear systems based on dual time stepping is applied. This method is faster than using Newton solvers. It was used successfully for solving Nonlinear CFD equations in the frequency domain. This new approach allows identifying the steady state of worn systems by integrating wear rate equations a on dual time scale. The dual time equations are integrated by an implicit scheme. Of the different orders tested, the first order scheme provided the best results.

Experimental measurements of out-of-plane vibrations of a simple blisk design using Blade Tip Timing and Scanning LDV measurement methods

The study of dynamic properties of rotating structures, such as bladed discs, can be conveniently done using simple bladed discs where the blades do not have staggering angles. Simplified design, although not truly representative of real structures, can be easy and economic to manufacture and, still, very helpful for studying specific dynamic properties. An example of this can be called as mass mistune blisk study. Experimental measurements of vibrations of bladed discs under rotating conditions can be performed using Scanning Laser Doppler Vibrometer (SLDV) systems. However, in the aerospace industry, the vibrations of complex bladed discs must be measured under operating conditions which are more hostile than laboratory simulations. The Blade Tip Timing (BTT) measurement method is a measurement technique, which can be used to measure vibrations of bladed discs of an engine aircraft under operating conditions. However, the BTT technique is ineffective when used with a flat bladed disc whose blade vibrations cannot be measured. This can be detrimental when the use of controlled dynamic parameters, such as those obtained from a simple bladed disc design, can improve the confidence for the validation of post-processing software.This paper presents a work about experimental measurements of a simple bladed disc design whose vibrations were measured synchronously by Scanning LDV and BTT measurement systems. A rotating test rig and its mechanical modifications for the installation of the BTT probes are introduced. Implications of rotating a specimen inconsistently are presented so as solutions to obtained constant revolving speed. The experimental comparisons of forced response vibrations measured synchronously at one blade are presented and explained.

A Comprehensive Procedure to Estimate the Probability of Extreme Vibration Levels Due to Mistuning

A new procedure is developed to find the probabilities of extremely high amplification factors in mistuned bladed disk vibration levels, typical of events which occur rarely. While a rough estimate can be made by curve-fitting the distribution function generated in a Monte Carlo simulation, the procedure presented here can determine a much more accurate upper bound and the probabilities of amplification factors near to that bound. The procedure comprises an optimization analysis based on the conjugate gradient method and a stochastic simulation using the importance sampling method. Two examples are provided to illustrate the efficiency of the procedure, which can be 2 or 3 orders of magnitude more efficient than Monte Carlo simulations.

Applications of continuous tracking SLDV measurement methods to axially symmetric rotating structures using different excitation methods

This paper presents the latest developments of the continuous-scanning laser Doppler vibrometry (C-SLDV) measurement methods applied to rotating axially symmetric structures such as bladed discs. Measurement of vibrations of rotating structures are still difficult to perform. The main reason is due to the limitations of conventional transducers used for capturing the vibrations, these are often unable to measure and/or transfer a good quality signal remotely. The LDV became an ideal candidate to replace/aid such transducers, but only the introduction of two scanning mirrors in front of a laser head produced a step forward in measurements of vibrations of rotating targets; SLDV systems are now widely used both in industry and academic communities. This work shows how a commercial SLDV system can be used for measurement of vibrations of bladed discs under rotating conditions. The features of the scanning head, which was not modified for this study, were exploited up to the limit to achieve the synchronization between the scanner and a rotating target. The simplest tracking technique is performed when a circle-line scan pattern is synchronized with the rotating bladed disc so as to produce the point tracking measurement method. The extension of such a discrete measurement method to a full-field one is made when the laser beam is capable of tracing continuously either a line (LineScan tracking) or an area (AreaScan tracking) over the surface of the rotating structure. The development of tracking C-SLDV measurement techniques is achieved by the use of the traditional excitation methods and the development of a new excitation system, which will bring the excitation and the measurement to be in the same rotating frame of reference. Several experimental results are provided to illustrate the use of such techniques in turbomachinery industries.