Prediction Of Forced Response Research Articles

Efficient numerical prediction of blade forced response under inlet distortion

The compressor blade in an aircraft gas turbine engine may suffer from forced vibration problems under the unsteady aerodynamic force due to inlet distortion. This paper develops an efficient numerical method combination to predict the forced response. The computational fluid dynamics (CFD) method is combined with the Fourier transform method to simulate the unsteady flow field under inlet distortion. Then, the forced response of the blade is obtained by the frequency-domain method based on the modal force. The results show that the developed method combination improves the unsteady flow calculation efficiency by 20 times and the response calculation efficiency by 48 times at the cost of less than 5% accuracy compared with the traditional method combination, including the unsteady flow field calculation method based on the three-row full-annulus model and the transient analysis method for calculating the forced response. Finally, the developed method combination is used to analyze the forced response of the blade under different inlet total pressure distributions. The results indicate that the forced response can be effectively decreased by decreasing the total pressure difference and increasing the sector angle difference between the high-pressure and low-pressure areas.

Forced response in a multi-stage aeroderivative axial compressor with acoustic excitations: Method validation with field data

This paper presents the application of a forced response prediction process to an industrial aeroderivative axial compressor and its validation against experimental data. The forced response process is based on a dedicated strategy to decompose the overall unsteady aerodynamic forcing coming from URANS analyses, and an improved use of the interference diagram to detect additional acoustic forcing in the entire operating range. The identification of potential resonance conditions, or crossings, in the interference diagram is based on the assumptions of a cyclic symmetry tuned structure and a forcing function having an aliased engine order driven only by the airfoil count of the rotor/stator interactions of interest. Despite this, it is not rare that unexpected resonances appear during compressor validation tests, often calling for painful and time-consuming redesigns. This is witnessed by the unsteady analysis of an 11-stage axial compressor in which the spatial content of engine order forcing is influenced also by further rotor/stator interactions by means of Tyler-Sofrin acoustic modes that cause the onset of additional crossing conditions.The theory based on time and space decomposition is validated with full 3D unsteady forced response simulations of the entire multistage compressor. Numerical results, obtained by an in-house modal work approach, are compared with experimental data focusing on two resonances, the first of which can be detected by the classical interference diagram, while the second one is justified only by the improved use that considers the acoustic excitations. In both cases, the predicted blade responses are in good agreement with measurements.The theory associated with the generation mechanism of the additional nodal diameters is further explained using the results of dedicated numerical experiments conducted by changing the relative stator and rotor airfoil count on a reduced compressor domain.

A test rig for the full characterization of the dynamics of shrouded turbine blades

Highly stressed forced vibrations experienced by Low Pressure Turbine (LPT) blades during their operation can result in high cycle fatigue (HCF) which can ultimately lead to failure. To avoid this occurrence, the vibration amplitudes must be anticipated and reduced. This is generally done by employing friction damping devices like under-platform dampers, shrouds and snubbers. In case of blades with shrouds at their tips, the adjacent shrouded blades are coupled to each other and three-dimensional periodic shroud contact forces acting on shrouds significantly influence the vibration amplitudes as energy is dissipated due to friction. Hence, for the experimental validation of the numerical contact models that are used for the prediction of nonlinear forced response of shrouded blades, it is equally necessary to measure the contact forces acting at the shrouds. Moreover, present-day requirement of more accurate and detailed models requires thorough and comprehensive experimental validation calling for the measurement of three-dimensional shroud contact forces from purposely designed test rigs.This study presents the design, development and operation of an experimental test rig that allows full characterization of the dynamics of shrouded turbine blade, i.e., simultaneous measurement of three-dimensional shroud contact forces and shrouded blade forced response. The starting point was the design of the test rig components based on the design requirements. This is followed by the description of the major components of the proposed test rig i.e., the three-dimensional contact force measurement system and the torque screw mechanism. In the subsequent section, the details of the experimental setup to measure the forced response and three-dimensional shroud contact forces simultaneously are highlighted as the test campaign is performed for different normal preloads and excitation force levels. Consequently, the effects of the variation in the normal preload and excitation force on the measured response and shroud contact forces are discussed. Finally, it is shown how the results establish the efficacy of the proposed test rig in providing a comprehensive depiction of the dynamic response of the shrouded blade and 3D shroud contact forces that will result in more accurate and reliable experimental validation of numerical tools.

Application of Machine Learning to Forced Response Predictions of an Industrial Axial Compressor Rotor Blade

Abstract Machine learning has gone way beyond a ground-breaking technology a few decades ago to now taken for granted in many day-to-day activities. It is now providing new ways for manufacturing, assembling, operating, monitoring, and maintaining products. Typical application areas include performance optimization, quality improvements, fault detection and predictive maintenance. In this paper application of machine learning algorithms to forced response prediction during the design and analysis of a typical gas turbine compressor blade is reported. The forced response prediction process typically involves utilizing harmonic or time domain computational fluid dynamics (CFD) analyses to compute the forcing and the aero damping, to calculate reserve factors that represent the high cycle fatigue life of the blade. This time-consuming process is generally limited to the later phases of the design cycle and can lead to hundreds of calculations if one must address all the resonances in a typical twin shaft running range. A neural network trained using historical data is used to directly predict the reserve factor with high confidence without the need for costlier high fidelity CFD by using just the finite elements (FE) predicted parameters. This allows to perform high-fidelity aero-mechanical assessment at an early stage in the design process. Further, application of image recognition using a convoluted neural network to aid in the identification of FE predicted Modeshapes is also demonstrated, which can also improve the quality of the reserve factor predictions.

Practical Method for Burner Staging Turbine Forced Response Evaluation

Abstract Decarbonization and sustainability efforts challenge gas turbine engineers to come up with creative strategies for reduction of emissions and efficiency increase over the whole operating range. Burner staging at part loads presents a flexible solution to achieve these goals through selective burner deactivation. Shutting off burners could also be required for combustion of increased H2 content at some conditions. Burner staging will create circumferential unevenness with patterns of hot and cold streaks that could excite blade rows through the entire turbine. This paper presents a parametric method for annular combustor staging patterns' profile generation intended for use for forced response predictions from a limited number of combustor computational fluid dynamics (CFD) calculations while keeping the key phenomenological features. Two cases with burner staging turbine inlet temperature (TIT) distributions are considered and compared to a base case with uniform temperature distribution. The unsteady aerodynamic forcing was obtained from full wheel time marching unsteady CFD calculations. The results show that the hot streaks generate important and noticeable excitation sources. Additionally, the results show that the pattern generator could be used extensively before and after the unsteady calculations' phase to minimize the excitation levels and the computational load.

Design and Calibration of a Tri-Directional Contact Force Measurement System

In low pressure turbine stages, adjacent blades are coupled to each other at their tip by covers, called shrouds. Three-dimensional periodic contact forces at shrouds strongly affect the blade vibration level as energy is dissipated by friction. To validate contact models developed for the prediction of nonlinear forced response of shrouded blades, direct contact force measurement during dynamic tests is mandatory. In case of shrouded blades, the existing unidirectional and bi-directional contact force measurement methods need to be improved and extended to a tri-directional measurement of shroud contact forces for a comprehensive and more reliable validation of the shroud contact models. This demands an accurate and robust measurement solution that is compatible with the nature and orientation of the contact forces at blade shrouds. This study presents a cost effective and adaptable tri-directional force measurement system to measure static and dynamic contact forces simultaneously in three directions at blade shrouds during forced response tests. The system is based on three orthogonal force transducers connected to a reference block that will eventually be put in contact with the blade shroud in the test rig. A calibration process is outlined to define a decoupling matrix and its subsequent validation is demonstrated in order to evaluate the effectiveness of the measurement system to measure the actual contact forces acting on the contact.

A modal based reduction technique for wide loose interfaces and application to a turbine stator

Aim of this research is the prediction of the non-linear forced response of a stator bladed disk for aeronautical applications in the presence of friction contacts at the hook joints (casing – vane segment contact). Due to the large extension of the contact interfaces the use of classic node-to-node contact elements based on the Coulomb friction law becomes cumbersome. Although such approach appears convenient for problems involving small contact regions, it is computationally expensive when large contact interfaces with refined meshes are present between components. In this research, a novel modal interface reduction method is combined with a layer of Jenkins contact elements in order to efficiently predict the effects of friction on the non-linear forced response of the stator bladed disk. The wide rail/hook contact regions are therefore suitable for the application of the presented technique. The goodness of the proposed methodology is quantified both in terms of accuracy and time costs savings on the calculation of non-linear forced responses by exploiting cyclic symmetry hypotheses and the Harmonic Balance Method.

Rotor–Stator Interactions in a 2.5-Stage Axial Compressor—Part II: Impact of Aerodynamic Modeling on Forced Response

Abstract The main objective of this study is the validation of numerical forced response predictions through experimental blade vibration measurements for higher order modes of a blade-integrated disk (blisk). To this end, a linearized and a nonlinear frequency domain CFD methods are used, as well as a tip timing measurement system. The focus is on the blade excitation by downstream vanes, in particular, because this study shows that the correct prediction of acoustic modes is of key importance in this case. The analysis of these modes is presented, both experimentally and numerically, in Part I of this publication. The grid independence study for the aerodynamic work on the blade surface conducted in this part shows a possible prediction uncertainty of more than 100% when a coarse grid is chosen. For the validation of the numerical setup, a study was performed using different turbulence and transition models. The results are compared to the measured performance map, to a 2D field traverse conducted with a pneumatic probe, and to data gained by unsteady pressure sensors mounted in the casing of the compressor. Flow features relevant for the prediction of blade stresses are best represented using the SST turbulence model in combination with the γ–ReΘ transition model. Nonlinear simulations with this setup are able to predict the blade stresses due to downstream excitation with an average difference of 23% compared to tip timing measurements. Single row linearized CFD methods have shown to be incapable of making a correct stress prediction when acoustic modes form a major part of the exciting mechanisms. In summary, this two-part publication proves the importance of acoustic rotor–stator interactions for blade vibrational stresses excited by downstream vanes in a state-of-the-art high-pressure compressor.

On the vibrational dynamics of rotating thin-walled cylinders: A theoretical and experimental study utilizing active magnetic bearings

This paper describes a theoretical and experimental study to establish the vibrational dynamics of a rotating thin-walled cylinder with radial bearing supports. The main focus is on the prediction of forced response, and the variation in response behaviour with rotational speed. A shell theory analysis is shown to provide a very complete description of rotordynamic behaviour that predicts various types of natural mode for free vibration. These include in-surface torsional and extensional modes, out-of-surface wall bending modes, as well as the classical beam bending modes exhibited by long flexible rotors. For exact solution of the free vibration problem, the coupled eigenproblems derived from the continuum equations and boundary constraints can be solved numerically. This approach can be applied for any given rotational speed. To avoid having to solve the equations repeatedly, an alternative model is formulated based on the zero-speed mode-shapes which has a simple parametric dependency on rotational speed. The method is applied to the dynamic modelling of an experimental system comprising a 0.8 m long steel rotor with outer diameter of 0.166 m and wall-thickness of 3.1 mm supported by two radial active magnetic bearings. The dynamic behaviour of the system is identified by frequency response testing at different rotational speeds, where excitation forces are applied through the bearings. The results confirm the accuracy and applicability of the developed shell theory model for practical use in rotordynamic prediction and analysis.

Analytic prediction of isolated forced response curves from spectral submanifolds

We show how spectral submanifold (SSM) theory can be used to provide analytic predictions for the response of periodically forced multi-degree-of-freedom mechanical systems. These predictions include an explicit criterion for the existence of isolated forced responses that will generally be missed by numerical continuation techniques. Our analytic predictions can be refined to arbitrary precision via an algorithm that does not require the numerical solutions of the mechanical system. We illustrate all these results on low- and high-dimensional nonlinear vibration problems. We find that our SSM-based forced response predictions remain accurate in high-dimensional systems, in which numerical continuation of the periodic response is becoming computationally expensive.

Model update and validation of a mistuned high-pressure compressor blisk

ABSTRACTIn order to prepare an advanced 4-stage high-pressure compressor rig test campaign, details regarding both accomplishment and analysis of preliminary experiments are provided in this paper. The superior objective of the research project is to contribute to a reliable but simultaneously less conservative design of future high pressure blade integrated disks (blisk). It is planned to achieve trend-setting advances based on a close combination of both numerical and experimental analyses. The analyses are focused on the second rotor of this research compressor, which is the only one being manufactured as blisk. The comprehensive test program is addressing both surge and forced response analyses e.g. caused by low engine order excitation. Among others the interaction of aeroelastics and blade mistuning is demanding attention in this regard. That is why structural models are needed, allowing for an accurate forced response prediction close to reality. Furthermore, these models are required to support the assessment of blade tip timing (BTT) data gathered in the rig tests and strain gauge (s/g) data as well. To gain the maximum information regarding the correlation between BTT data, s/g-data and pressure gauge data, every blade of the second stage rotor (28 blades) is applied with s/g. However, it is well known that s/g on blades can contribute additional mistuning that had to be considered upon updating structural models.Due to the relevance of mistuning, efforts are made for its accurate experimental determination. Blade-by-blade impact tests according to a patented approach are used for this purpose. From the research point of view, it is most interesting to determine both the effect s/g-instrumentation and assembling the compressor stages on blade frequency mistuning. That is why experimental mistuning tests carried out immediately after manufacturing the blisk are repeated twice, namely, after s/g instrumentation and after assembling. To complete the pre-test program, the pure mechanical damping and modal damping ratios dependent on the ambient pressure are experimentally determined inside a pressure vessel. Subsequently the mistuning data gained before is used for updating subset of nominal system mode (SNM) models. Aerodynamic influence coefficients (AICs) are implemented to take aeroelastic interaction into account for forced response analyses. Within a comparison of different models, it is shown for the fundamental flap mode (1F) that the s/g instrumentation significantly affects the forced response, whereas the impact of assembling the compressor plays a minor role.

Forced Response Prediction of Turbine Blades with Flexible Dampers: The Impact of Engineering Modelling Choices

This paper focuses on flexible friction dampers (or “strips”) mounted on the underside of adjacent turbine blade platforms for sealing and damping purposes. A key parameter to ensure a robust and trustworthy design is the correct prediction of the maximum frequency shift induced by the strip damper coupling adjacent blades. While this topic has been extensively addressed on rigid friction dampers, both experimentally and numerically, no such investigation is available as far as flexible dampers are concerned. This paper builds on the authors’ prior experience with rigid dampers to investigate the peculiarities and challenges of a robust dynamic model of blade-strips systems. The starting point is a numerical tool implementing state-of-the-art techniques for the efficient solution of the nonlinear equations, e.g., multi-harmonic balance method with coupled static solution and state-of-the-art contact elements. The full step-by-step modelling process is here retraced and upgraded to take into account the damper flexibility: for each step, key modelling choices (e.g., mesh size, master nodes selection, contact parameters) which may affect the predicted response are addressed. The outcome is a series of guidelines which will help the designer assign numerical predictions the proper level of trust and outline a much-needed experimental campaign.

Mistuned Higher-Order Mode Forced Response of an Embedded Compressor Rotor—Part II: Mistuned Forced Response Prediction

This paper is the second part of a two-part paper that presents a comprehensive study of the higher-order mode (HOM) mistuned forced response of an embedded rotor blisk in a multistage axial research compressor. The resonant response of the second-stage rotor (R2) in its first chordwise bending (1CWB) mode due to the second harmonic of the periodic passing of its neighboring stators (S1 and S2) is investigated computationally and experimentally at three steady loading conditions in the Purdue three-stage compressor research facility. A nonintrusive stress measurement system (NSMS, or blade tip-timing) is used to measure the blade vibration. Two reduced-order mistuning models of different levels of fidelity are used, namely, the fundamental mistuning model (FMM) and the component mode mistuning (CMM), to predict the response. Although several modes in the 1CWB modal family appear in frequency veering and high modal density regions, they do not heavily participate in the response such that very similar results are produced by the FMM and the CMM models of different sizes. A significant response amplification factor of 1.5–2.0 is both measured and predicted, which is on the same order of magnitude of what was commonly reported for low-frequency modes. In this study, a good agreement between predictions and measurements is achieved for the deterministic analysis. This is complemented by a sensitivity analysis which shows that the mistuned system is highly sensitive to the discrepancies in the experimentally determined blade frequency mistuning.

A Taylor Series Expansion Approach for Nonlinear Blade Forced Response Prediction Considering Variable Rotational Speed

In the field of turbomachinery, great efforts are made to enhance computational tools to obtain reliable predictions of the vibrational behavior of friction-damped bladed disks. As a trade-off between computational burden and level of simplification, numerous methods were developed to reduce the nonlinear systems dimension. Using component mode synthesis methods (CMS), one is capable to describe the systems motion by interface and modal coordinates. Subsequently or alternatively, the dynamic compliance matrix can be evaluated efficiently by means of modal superposition to avoid the inversion of the dynamic stiffness matrix. Only the equations corresponding to the degrees-of-freedom (DOF) subject to localized nonlinear contact forces need to be solved simultaneously, whereas the solution of the linear DOF is obtained by exploiting the algebraic character of the set of equations. In this paper, an approach is presented to account for rotational speed-dependent stiffness in the subset of nonlinear DOF without the need to re-evaluate the associated eigenvalue problem (EVP) when rotational speed is changed. This is done by means of a Taylor series expansion of the eigenvalues and eigenvectors used for the modal superposition to reconstruct the dynamic compliance matrix. In the context of forced response predictions of friction-damped blisks, the expansion is performed up to different order for a simplified blisk model with nonlinear contact interfaces. The results are compared to the solution obtained by direct evaluation of the EVP at selected rotational speeds and the solution when dynamic compliance matrix is built up by direct inversion of the dynamic stiffness matrix. Finally, the proposed methods computational performance is analyzed.

Enhanced wave and finite element method for wave propagation and forced response prediction in periodic piezoelectric structures

As a promising numerical tool of structural dynamics in mid- and high frequencies, the wave and finite element method (WFEM) is receiving increasingly attention and applications. In this paper, an enhanced WFEM has been developed with a reduced model and a new eigenvalue scheme. The reduced model is applicable for structures with piezoelectric shunts or local dampers; the new eigenvalue scheme can mitigate the ill-conditioning when the wave basis is calculated. The enhanced WFEM is applied to a thin-wall structure with periodically distributed piezoelectric materials (PZT). Both free wave characteristics and forced response are analyzed and the influences of the suggested enhancements are presented. It is shown that if the control factors are properly chosen, these enhancements can improve the accuracy while accelerating the calculation. Resulting from the complexity of the application, these enhancements are not optional but imperative.

Influence of the Tip Clearance on a Compressor Blade Aerodynamic Damping

The flutter CFD calculations conducted in industry generally use a single-row, single-passage approach with a minimum of detailing to reduce the computational time as much as possible. The question arises of what level of detailing is necessary for accurate aerodamping predictions. In this study, the influence of tip clearance size on the steady flowfield and aerodynamic damping is assessed for a high-pressure compressor second-stage rotor. The steady analyses are conducted on an isolated blade passage by imposing the experimental boundary conditions at the inlet and outlet of the passage. Two operating points are considered: 74 and 100% corrected speed. It is observed that, at the same pressure ratio, the mass flow decreases with a larger tip gap due to the tip clearance flow rolling into a vortex on the blade suction side. Then, unsteady analyses are conducted by enforcing the first torsion mode shape at the corresponding natural frequency. For all interblade phase angles, the same trend is observed: the damping first increases (stabilizing effect) with the tip clearance before reaching a maximum and dropping for the larger gaps. Such changes in damping would have a significant impact on both flutter and forced response predictions.

Mistuned Forced Response Predictions of an Embedded Rotor in a Multistage Compressor

This paper covers a comprehensive forced response analysis conducted on a multistage compressor and compared with the largest forced response experimental data set ever obtained in the field. The steady-state aerodynamic performance and stator wake predictions compare well with the experimental data, although losses are underestimated. Coupled and uncoupled unsteady simulations are conducted on the stator–rotor configuration. It is shown that the use of a decoupled method for forced response cannot yield accurate results for cases with strong inter-row interactions. The individual and combined contributions of the upstream and downstream stators are also assessed. The downstream stator is found to have a tremendous impact on the forced response predictions due to the constructive interactions of the two stator rows. Finally, predicted mistuned blade amplitudes are compared to mistuned experimental data. The average amplitudes match the experiments very well, while the maximum response amplitude is underestimated.

An Information-Theoretic Framework for Improving Imperfect Dynamical Predictions Via Multi-Model Ensemble Forecasts

This work focuses on elucidating issues related to an increasingly common technique of multi-model ensemble (MME) forecasting. The MME approach is aimed at improving the statistical accuracy of imperfect time-dependent predictions by combining information from a collection of reduced-order dynamical models. Despite some operational evidence in support of the MME strategy for mitigating the prediction error, the mathematical framework justifying this approach has been lacking. Here, this problem is considered within a probabilistic/stochastic framework which exploits tools from information theory to derive a set of criteria for improving probabilistic MME predictions relative to single-model predictions. The emphasis is on a systematic understanding of the benefits and limitations associated with the MME approach, on uncertainty quantification, and on the development of practical design principles for constructing an MME with improved predictive performance. The conditions for prediction improvement via the MME approach stem from the convexity of the relative entropy which is used here as a measure of the lack of information in the imperfect models relative to the resolved characteristics of the truth dynamics. It is also shown how practical guidelines for MME prediction improvement can be implemented in the context of forced response predictions from equilibrium with the help of the linear response theory utilizing the fluctuation–dissipation formulas at the unperturbed equilibrium. The general theoretical results are illustrated using exactly solvable stochastic non-Gaussian test models.

Dynamic response characteristics of dual flow-path integrally bladed rotors

New turbine engine designs requiring secondary flow compression often look to dual flow-path integrally bladed rotors (DFIBRs) since these stages have the ability to perform work on the secondary, or bypassed, flow-field. While analogous to traditional integrally bladed rotor stages, DFIBR designs have many differences that result in unique dynamic response characteristics that must be understood to avoid fatigue. This work investigates these characteristics using reduced-order models (ROMs) that incorporate mistuning through perturbations to blade frequencies. This work provides an alternative to computationally intensive geometric-mistuning approaches for DFIBRs by utilizing tuned blade mode reductions and substructure coupling in cyclic coordinates. Free and forced response results are compared to full finite element model (FEM) solutions to determine if any errors are related to the reduced-order model formulation reduction methods. It is shown that DFIBRs have many more frequency veering regions than their single flow-path integrally blade rotor (IBR) counterparts. Modal families are shown to transition between system, inner-blade, and outer-blade motion. Furthermore, findings illustrate that while mode localization of traditional IBRs is limited to a single or small subset of blades, DFIBRs can have modal energy localized to either an inner- or outer-blade set resulting in many blades responding above tuned levels. Lastly, ROM forced response predictions compare well to full FEM predictions for the two test cases shown.

A Reduced-Order Meshless Energy Model for the Vibrations of Mistuned Bladed Disks—Part I: Theoretical Basis

In this paper, a reduced order model for the vibrations of bladed disk assemblies was achieved. The system studied was a 3D annulus of shroudless, “custom-tailored,” mistuned blades attached to a flexible disk. Specifically, the annulus was modeled as a spectral-based “meshless” continuum structure utilizing only nodal data to describe the arbitrary volume in which the system's dynamical energy was minimized. An extended Ritz variational procedure was used to minimize this energy, subjected to constraints imposed by an assumed 3D displacement field of mathematically complete, orthonormal “blade-disk” polynomials multiplied by generalized coefficients. The coefficients were determined by constraining the polynomial series to satisfy the extended Ritz stationary equations and essential boundary conditions of the bladed disk. From this, the governing equations of motion were generated into their usual dynamical forms to calculate upper-bounds on the actual free and forced responses of bladed disks. No conventional finite elements and element connectivity or component substructuring data were needed. This paper, Part I, outlines the theoretical foundation of the present model, and through extensive Monte Carlo simulations, establishes the analytical basis, predictive accuracy, and re-analysis efficiency of the present technology in the prediction of 3D maximum response amplitude of mistuned bladed disks having increasing numbers of nodal diameter excitations. Further applications validating the 3D approach against conventional finite element procedures of free and forced response prediction of a mistuned Integrally-Bladed Rotor used in practice is presented in a companion paper, Part II (Fang, McGee, and El-Aini, 2013, “A Reduced-Order Meshless Energy Model for the Vibrations of Mistuned Bladed Disks—Part II: Finite Element Benchmark Comparisons, ASME J. Turbomach., to be published.