Validation of Forced Response Methods forTurbine Blades

Author:Hugo Hultman hhultman@kth.se

Supervisors:Clas AnderssonStaffan BrodinPieter GrothBjorn Laumert

clas.andersson@gknaerospace.comstaffan.brodin@gknaerospace.com

pieter.groth@gknaerospace.combjorn.laumert@energy.kth.se

Date:2015-04-20

1 Summary

Forced response is an important part of turbine blade structural integrityanalysis. Failure to identify possible resonances in the blades can lead toturbine failure, and reliable and accurate methods are required to avoid this.Here, a group of methods and different approaches for forced response areevaluated using a resonance in a reference turbine for testing. An analysiswas carried out using standard methods used at GKN Aerospace comparingdifferent options for CFD input, FFT, CFD to FE mesh mapping, and FEmodel coarseness. The same case was also run in AROMA , a forced responsetool being developed at KTH. The study revealed a difference in pressuredistribution and maxima between ANSYS CFX and VOLSOL CFD input,however the forced response results where roughly equal. The change in theFE model mesh density gave a notable but ultimately acceptable reduction ofaccuracy for forced response when comparing the displacement amplitude. Alarge difference was found between the FFT and mapping procedures carriedout for this report and the methods used during a previous study at GKNAerospace. AROMA produced promising results, but the tool needs to bemade more robust and able to handle larger FE models in order to becomeuseful for industry applications.

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Contents

1 Summary 1

2 Nomenclature 4

3 Introduction 5

4 Basic Theory 64.1 Forced Response . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 Aerodynamic Loading . . . . . . . . . . . . . . . . . . . . . . 94.4 Disc Eigenmode Excitation . . . . . . . . . . . . . . . . . . . . 11

5 State of The Art 145.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Aerodynamic Loading . . . . . . . . . . . . . . . . . . . . . . 15

5.2.1 VOLSOL . . . . . . . . . . . . . . . . . . . . . . . . . 155.2.2 ANSYS CFX . . . . . . . . . . . . . . . . . . . . . . . 15

5.3 Forced Response Analysis . . . . . . . . . . . . . . . . . . . . 155.3.1 ANSYS Mechanical . . . . . . . . . . . . . . . . . . . 155.3.2 The AROMA Code . . . . . . . . . . . . . . . . . . . . 15

6 Method 176.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.2 Inputs and Models . . . . . . . . . . . . . . . . . . . . . . . . 186.3 CFD Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.4 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 216.5 Mapping of Loads . . . . . . . . . . . . . . . . . . . . . . . . . 236.6 Forced Response in ANSYS Mechanical . . . . . . . . . . . . 24

6.6.1 Prestress Conditions . . . . . . . . . . . . . . . . . . . 256.6.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . 256.6.3 Harmonic analysis . . . . . . . . . . . . . . . . . . . . 25

6.7 Forced Response in AROMA . . . . . . . . . . . . . . . . . . 266.7.1 Pre-processing . . . . . . . . . . . . . . . . . . . . . . . 266.7.2 Initiation and Settings . . . . . . . . . . . . . . . . . . 27

7 Results 297.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.2 Specific Comparisons . . . . . . . . . . . . . . . . . . . . . . . 29

7.2.1 ANSYS CFX and VOLSOL . . . . . . . . . . . . . . . 297.2.2 Fine and Coarse Mesh . . . . . . . . . . . . . . . . . . 31

7.2.3 FFT and Mapping . . . . . . . . . . . . . . . . . . . . 327.2.4 AROMA Settings . . . . . . . . . . . . . . . . . . . . . 327.2.5 ANSYS Mechanical and AROMA . . . . . . . . . . . 33

7.3 Test Data Comparison . . . . . . . . . . . . . . . . . . . . . . 34

8 Conclusions 358.1 On Result Interpretation . . . . . . . . . . . . . . . . . . . . . 358.2 CFD Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.3 Mesh Density Reduction . . . . . . . . . . . . . . . . . . . . . 358.4 FFT and CFD to FE Mesh Mapping . . . . . . . . . . . . . . 368.5 Test Data Comparison . . . . . . . . . . . . . . . . . . . . . . 368.6 AROMA Evaluation . . . . . . . . . . . . . . . . . . . . . . . 37

8.6.1 Analysis Results . . . . . . . . . . . . . . . . . . . . . 378.6.2 Usability . . . . . . . . . . . . . . . . . . . . . . . . . . 378.6.3 Verdict . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

9 Acknowledgments 39

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2 Nomenclature

2S3AROMACFDCMSDOFEOFEFFTHCFROMRTZZENF

Stator 3 2nd HarmonicAeroelastic Reduced Order Modeling AnalysesComputational Fluid DynamicsComponent Mode SynthesisDegree Of Freedom(s)Engine OrderFinite ElementFast Fourier TransformHigh Cycle FatigueReduced Order ModelingReference TurbineZig-zag Engine Order Nodal Diameter Frequency

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3 Introduction

Forced response analysis is a vital part of turbomachinery design. Due tothe periodic nature of the aerodynamic loading in a spinning structure, greatcare must be taken to ensure that no dangerous resonances will occur duringoperation of the turbine. Simulation methods are important for identify-ing frequencies perceptible to resonance, allowing engineers to address theseproblems early in the design process.

The purpose of this report is to evaluate and seek to validate a group ofdifferent methods for forced response analysis. The work was carried out atGKN Aerospace under the supervision of engineers experienced with forcedresponse at the company. All test cases are based on a turbine project atGKN Aerospace, in this report referred to as Reference Turbine (RT). Theturbopump has a blisk-type rotor forged in a single piece, meaning that itis especially vulnerable to resonances due to its low damping. A forced re-sponse analysis has previously been done at the company, but the need foraccurate tools for forced response means that the field is constantly evolving.This calls for an up to date validation of current methods.

Another objective of the validation work is to evaluate a new forced re-sponse tool called AROMA and currently in development at KTH. By con-ducting analyses in AROMA in parallel to the more traditional methods usedat GKN Aerospace, results can be compared in order to evaluate the codeand provide feedback to its development.

The report is divided into five main parts: basic theory, state of the art,method, results, and conclusions. First, the mathematical background toforced response is briefly explained in a theory chapter to give the reader abasic understanding of the phenomena. The next section outlines the analysisapproach and addresses the current state of the field, discussing the differentengineering softwares involved in the methods being evaluated. The methodsection explains in detail how the analyses were carried out, outlining thedifferent steps and settings used. Finally the results are presented with rele-vant comparisons, followed by a section with conclusions.

NOTE: In order to comply with GKN Aerospace confidentiality policy,this report has been edited to hide sensitive information. Figures containingmodels have been distorted to hide exact dimensions of the rotor, and allplots have been rescaled.

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4 Basic Theory

4.1 Forced Response

In order to gain an understanding of the vibration phenomena that can occurin a spinning turbine disc, it is convenient to start by looking at a singledegree of freedom (DOF) system excited by a time-varying force F(t). Amass m is mounted on a spring with stiffness k and damping coefficient caccording to Figure 1.

Figure 1: Model for a damped single DOF system with time-varying excita-tion force.

As the system is excited by the time-varying force F(t) it will gain somemotion u(t) determined by the equation of motion (EOM)

mu(t) + cu(t) + ku(t) = F (t). (1)

To analyze the system, the second order non-homogeneous differential equa-tion (1) must be solved for u(t). This yields a solution consisting of a com-plementary solution uc(t) and a particular solution up(t), written as

u(t) = uc(t) + up(t), (2)

where uc(t) is a solution to the homogeneous form of equation (1),

mu(t) + cu(t) + ku(t) = 0. (3)

The particular solution up(t) is based on the right hand side term of thenon-homogeneous equation (1), in this case the excitation force F(t).

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In the case of a differential equation describing a vibrating damped systemsuch as the one described in Figure 1, a physical interpretation of the par-ticular and complementary solutions can be made. Here the complementarysolution describes the motion of a freely vibrating system, which is simplygoverned by its initial condition and where the oscillations will slowly die outdue to the damping in the system. This can be observed when a mass on aspring is pulled from its equilibrium position and then let go. The particularsolution on the other hand describes the part of the motion u(t) which is di-rectly caused by the excitation force F(t), and will thus describe an ongoingmotion which will continue while the system is being excited by an externaltime-varying force.

By further applying these results to the case of an elastic body surroundedby a moving fluid such as an airplane wing, propeller blade, or certainly aturbine disc, more specific interpretations can be made. Here the comple-mentary solution is related to the field of aeroelasticity, where phenomenasuch as flutter are studied. Flutter can occur even if the flow is completelystationary when a small initial perturbation causes vibrations in the struc-ture which are amplified by the surrounding flow. Aeroelastic instabilitiessuch as flutter are outside the scope of this report, and will not be discussedin more detail.

Looking again at the body surrounded by a moving fluid, the particularsolution to the equations of motion will describe a motion excited purely byturbulence and disturbances in the flow. Having no connection to the initialcondition of the system, the motion is determined by the excitation force,damping, mass and stiffness parameters. This is called forced response.

4.2 Modal Analysis

When looking at more complex systems, the single DOF model in Figure 1quickly becomes obsolete and more DOFs are added to simulate the elasticityof the body being analyzed. This is modeled as a network of masses connectedby unique stiffness and damping parameters, arranged in matrices to form amultiple DOF system of equations

Mu(t) + Cu(t) + Ku(t) = F (t) (4)

where the vectors F(t) and u(t) describe the excitation force applied at eachnode and its corresponding motion. The system characteristics are describedby mass, stiffness, and damping matrices denoted M, K, and C.

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Solving the system of differential equations formed by the EOM (4) re-quires a lot of time and computational power. In this study, an alternative tosolving the complete system of differential equations is to rewrite the EOMbased on its undamped eigenmodes and eigenfrequencies. For an undampedsystem, the eigenfrequencies and eigenmodes of a system are determined byfinding the nontrivial solutions to the eigenvalue problem

[K − ω2M ]U = 0 (5)

where ω is an eigenfrequency and U describes a modeshape. The eigenmodematrix has a special property that allows the mass and stiffness matrices tobe diagonalized as

UTMU = diag(m1,m2, ...,mn) (6)

UTKU = diag(k1, k2, ..., kn). (7)

By inserting the coordinate transformation for the deformation vector

u(t) = Uq(t), (8)

the undamped EOM can be rewritten by multiplying with UT giving

UTMUq(t) + UTKUq(t) = UTF (t). (9)

Because of the orthogonality of the modes in (6), the rewritten form (9) givesN uncoupled differential equations

mnqn(t) + knqn(t) = Qn(t) (10)

where mn(t), kn(t) and Qn(t) are the modal mass, stiffness, and force cor-responding to mode n. This allows the system to be analyzed based onthe behavior of its modes, and the physical deformation of the structure isfound by summarizing the modal deformation from each mode applied to itscorresponding modeshape giving

u(t) =N∑r=1

U r(t)qr(t). (11)

While the above assumes no damping in the system, it is generally desirableto introduce some form of damping force modeling. Damping is difficult tomodel, and different approaches are used depending on the structure andavailable computing capacity. For the analysis relevant to this report, linear

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viscous damping is used.

Damping can be described as a force acting in the opposite direction ofthe motion and is generally introduced as

Mu(t) + Ku(t) = F (t) + FD(t). (12)

Linear viscous damping is common in vibration analysis and means that thedamping FD(t) is modeled as a force proportional to the velocity as

FD(t) = −Cu(t). (13)

In the general case, a real valued matrix of eigenmodes U obtained from(5) cannot diagonalize the damping matrix C as in (6). In this study, theassumption of modal damping is used. This attributes a damping coefficientcn to each mode to be used in the uncoupled equations (10).

The damping coefficient is usually based on material or test data and isoften calculated based on the critical damping cnk. Critical damping is anamount of damping that constitutes the limit between an oscillating systemand one that returns directly to the equilibrium state. If the damping is belowcritical damping, the system is underdamped and will oscillate back and forthwith decreasing amplitude until the vibration dissipates. If the dampingis higher than critical damping, the system is overdamped and will returndirectly to its equilibrium state, however at a slower rate than a criticallydamped system. A damping property is often given as a damping ratio ζ.An underdamped system has a damping ratio ζ < 1 while an overdampedsystem has ζ > 1, allowing for a quick characterization of damping behavior.Using material and testing data for the structure, the modal damping cn canbe calculated as

cn = ζncnk (14)

using given modal damping ratio ζn. Damping can now be introduced in theuncoupled modal differential equations (10) as

mnqn(t) + cnqn(t) + knqn(t) = Qn(t), (15)

allowing for a complete vibration analysis of the system through the synthesisequation (11).

4.3 Aerodynamic Loading

In a turbine structure, the main source of excitation forces is aerodynamicloading caused by pressure variations surrounding the different blade rows.

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As the turbine disc rotates, the pressure distributions over the blade surfaceswill vary periodically. This results in harmonic time-varying loads where thefrequencies mainly depend on rotation speed and the number of blades andvanes on the rotors and stators. Figure 2 shows an example of how pressuredistributions can vary over time in a basic rotor-stator setup. The influencefrom the rotor blades on the stator blade can be clearly seen, here resultingin a harmonic stator excitation with a period equal to the passing time of arotor blade.

(a) Minimum pressure (b) Maximum pressure

Figure 2: Plots created in ANSYS CFX showing pressure distributions onrotor and stator blades at different stages of a blade passage

Examining the pressure over the rotor blades reveals that excitations alsotravel upstream, resulting in a slight pressure fluctuation on the rotor bladesas well. In more complex turbine structures, multiple stages with varyingnumbers of blades and vanes cause excitation of various frequencies. Theseare sorted under different engine orders. In a rotating structure, an engineorder signifies how many times a periodic excitation will occur during onefull revolution. An excitation caused by a disc with 30 blades has an engineorder of 30, resulting in an excitation frequency equal to 30 ∗ Ω Hz where Ωis the disc rotation speed in revolutions per second.

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In order to identify excitation frequencies of the aerodynamic loading, afourier decomposition of the time history of the blade loads is required. Usingcomputational fluid dynamics (CFD) software, a vector containing pressurevariation over time can be extracted. When an fast fourier transform (FFT)is applied to the time signal, the frequency spectrum can be analyzed toidentify the frequencies with high loading. By converting the frequencies toengine orders using the disc rotation speed Ω, it is also possible to deducewhat sources are causing the excitations.

4.4 Disc Eigenmode Excitation

When modal analysis is applied to a disc, the modes can be characterized byNodal diameters and Nodal circles. These manifest as diameters and circleson the disc that will not oscillate when their mode is being excited. When thedisc is vibrating, each mode forms a wave with frequency ωn and a certainnumber of nodes corresponding to the nodal diameter. Figure 3 shows avisualization of a single mode in a turbine disc having a nodal diameter of 3.

Figure 3: A modeshape of a turbine disc with nodal diameter 3

As the disc rotates, the forces acting on it will sweep across the discsurface with a rotation speed Ω depending on the discs own rotation speedas well as the excitation source. As stated earlier, this generates an excitationfrequency n∗Ω where n is the engine order. It can be shown that a resonancewill occur when

ωn = n ∗ Ω. (16)

In this case, the shapes of the vibration mode and the periodic excitationare perfectly synchronized and the vibration is reinforced, causing resonancein the structure. In order for the excitation and vibration pattern to line

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up, the engine order of the excitation must coincide with the nodal diameterof the eigenmode. If the excitation frequency of an engine order reaches aneigenfrequency with a corresponding nodal diameter when close to the designrotation speed of the disc, the structure will be susceptible to resonance.

When dealing with a rotationally periodic structure, there is an upperlimit to the perceivable nodal diameter. The disc will consist of a certainnumber of identical sections that make up the full structure, where the num-ber of sections corresponds to the number of blades. Having a nodal diameterND so that

ND ≥ N/2 (17)

where N is the blade count will result in aliasing of the signal as the cyclicsections become fewer than the amount of vibration nodes. This causes thesignal to fold at ND = N/2, and frequencies with higher nodal diameter willbe represented according to Figure 4, showing an example with 20 blades.As a result, all engine orders including the higher ones will excite modes withnodal diameters in a span limited by the number of blades.

Figure 4: Plot showing the aliasing effect on nodal diameters ND > N/2 ina rotor with 20 blades

The resulting plot shows where resonances could potentially occur, con-necting each engine order to a corresponding nodal diameter. The analysis isusually focused on a set of specific engine orders that represent blade countson the turbine disc and their overtones, found through an FFT analysis of

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the aerodynamic loading. In order to specify the frequencies that will beexcited by each engine order when the turbine is running, the y-axis of Fig-ure 4 is multiplied with the turbine design speed given in [rev/s], yielding aplot like the one in Figure 5. The example shows excitations coming fromEO 1-8, 16, and 32. Note that lines have been added above and below theline representing the design speed Ω. These specify an interval according tothe entire speed range plus a safety margin between Ωmin and Ωmax . Allfrequencies within this range are regarded as susceptible to excitation duringturbine operation.

Figure 5: Engine orders are converted to frequencies and a design speed rangeas well as relevant engine orders is marked

Using Figure 5, it is now possible to identify which frequency ranges oneach mode that fulfill the criteria (16) and having a nodal diameter compati-ble with the engine order. These are marked in the figure with black lines atEO 1-8, 16, and 32. Using results from the modal analysis, these frequencyranges can be checked for eigenfrequencies on each mode. When a frequencyis found within a range marked by a line, a marker is put on the line atthat frequency. The resulting plot is called a ZZENF diagram [6]. A ZZENFdiagram gives a clear presentation of how many potentially dangerous eigen-frequencies exist in the system and to which mode they belong.

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5 State of The Art

5.1 Overview

When doing a forced response analysis, both CFD and Structural analysisare required. The general outline of the process is given in Figure 6. Thefirst step is to do a modal analysis in order to find the eigenfrequencies andeigenmodes of the structure. By plotting the results in a ZZENF diagramand looking for modeshapes where nodal diameter and eigenfrequency corre-spond to the engine order and frequency of a harmonic excitation at operatingconditions, the resonances within the speed range of interest can be identified.

Figure 6: Basic workflow for forced response analysis

Before doing the forced response analysis, the aerodynamic loads whichcause the vibrations have to determined. This is where CFD comes in. Theoperating conditions of the system are given as input, with the results show-ing how the pressure over the surfaces will vary over time. When these resultsare ready, they are processed and applied in a forced response analysis.

The structural analysis that gives the forced response results is carriedout using some type of finite element (FE) analysis software. Note that forthis type of problem an uncoupled analysis is generally deemed sufficient,meaning that coupled fluid-structure interaction is not taken into account.

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5.2 Aerodynamic Loading

5.2.1 VOLSOL

VOLSOL is an in-house CFD solver at GKN Aerospace developed in 1994.It was widely used within the company up until 2013 when it was phasedout. Previously all forced response analyses at the company were carried outusing CFD input supplied by VOLSOL . Its successor, VOLSOL++, is stillin use but is not designed for sliding interfaces. This feature is required forturbine blade forced response analysis in order to handle unsteady bladerowinteraction in relation to the stators. After the phase-out of VOLSOL thecompany is instead looking at using ANSYS CFX for this type of analysis.

5.2.2 ANSYS CFX

The ANSYS CFX software is a general purpose CFD solver that was takeninto use at GKN Aerospace in 2005. Thanks to its ability to handle slidinginterfaces it can be used in forced response analyses, making it a suitablesuccessor to VOLSOL for this task. However, since the recent phase-out ofVOLSOL there is not a lot of test data for evaluating ANSYS CFX forcedresponse results. The version used in this report was ANSYS CFX 15.0.

5.3 Forced Response Analysis

5.3.1 ANSYS Mechanical

ANSYS Mechanical is an FE software widely used within the industry. Itis currently used as the standard approach at GKN Aerospace when dealingwith forced response. ANSYS Mechanical can handle this through cyclicsymmetry analysis where only a cyclic sector of the rotor is analyzed. A scriptis typically written to first perform a static analysis to apply static pressureand temperature loads followed by a modal analysis in the relevant frequencyspectrum and finally a harmonic analysis using mode-superposition. Theversion used in this report was ANSYS Mechanical 15.0.

5.3.2 The AROMA Code

Aeroelastic Reduced Order Model Analysis (AROMA ) began developmentat KTH in 2011. It is a tool built for assessing High Cycle Fatigue (HCF) dueto forced response in turbomachinery. The code is written in the MATLABscripting language. While forced response analysis in ANSYS Mechanicalused in this study relies on cyclic analysis, AROMA aims to use Reduced

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Order Modeling (ROM) to reduce computation cost while conserving an ac-ceptable level of accuracy. Reducing the number of DOFs is a simple way tospeed up computations, but needs to be done in a way that does not greatlyimpact forced response results. The order reduction was initially based onGuyan [2] reduction, which proved to be inadequate in terms of accuracy [1].

In order to improve accuracy, AROMA moved over to the ComponentMode Synthesis (CMS) approach. With CMS, components are reduced in-dividually before being assembled together. In AROMA , the rotor beinganalyzed is split into a blade and a disc component before reduction. Twodifferent CMS methods were integrated into the code: the Craig-Bamptonmethod and the Craig-Chang method.

Craig-Bampton is a fixed-interface technique, meaning that the reductionis performed through a transformation matrix based on static modes anddynamic modes. The Craig-Chang method is similar, but also uses rigidbody modes in the transformation matrix [3]. The transformation matricesare structured as

TCB =

[φik ψib

0 Ibb

](18)

TCC =

[φik ψib ψir

φbk ψbb ψbr

](19)

Where the first columns are dynamic modes, the second columns are thestatic modes, and the third column in the Craig-Chang matrix represents therigid body modes. Specific advantages and disadvantages of the two methodsin turbomachinery forced response still requires further testing. Due to theamount of RAM available to AROMA in MATLAB , the tool is limited in thenumber of DOFs it can handle. This means that highly detailed FE modelsmay need to be modified with a coarser mesh before use with AROMA .

The AROMA tool focuses on modal analysis and forced response. Aero-dynamic forces as well as stiffness matrix and mass matrix are required asinputs. The pre-processing needed for splitting the model into componentsis done in ANSYS Mechanical .

The CMS approach version of AROMA has previously been evaluatedwith a test case using the Craig-Chang reduction and showed promising re-sults for a high pressure turbine [4]. The AROMA code is still in developmentby Mauricio Gutierrez. The version of the code used in this work was pro-vided on February 4th 2015.

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6 Method

6.1 Overview

The analyses performed during this work was done in several steps. First,a CFD analysis was made using ANSYS CFX to calculate the aerodynamicloads. The output produced as well as a corresponding pressure file generatedin VOLSOL during a previous study [7] were then processed through an FFTand mapping process. These results were analysed in ANSYS Mechanical byapplying the loads on the structure and doing a harmonic forced responseanalysis.

The loads generated in ANSYS CFX were also mapped and applied toa coarser version of the model created for AROMA . A file containing loadsmapped and processed during the previous study was applied with the sameANSYS Mechanical settings to provide a reference. Then, the same case wasevaluated using the AROMA code, where the mapped ANSYS CFX loadswere applied to the coarse mesh using varying settings for comparison. Anoutline of the process can be viewed in Figure 7.

The objective was to investigate an a specific resonance in RT. The har-monic load is caused by the second harmonic of the excitation from theStator 3 bladerow (2S3). Through forced response analysis, the vibrationamplitudes were predicted in order to assess the strength of the resonance.Finally, results from different methods were compared to identify similaritiesand differences between them.

To verify results and evaluate potential gains in accuracy, certain resultswere also put through a High Cycle Fatigue (HCF) analysis. A similar anal-ysis done during the previous study was used for comparisons [7]. This way,the new results could be directly compared to both old analysis results andavailable test data.

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Figure 7: Analysis workflow outlining inputs, processes, and outputs

6.2 Inputs and Models

The analyses made in this report are based on a set of models that havebeen created and used in previous studies of the turbine. This allowed fora relatively effective work process as no time has been spent defining newstructural or material data. It also makes it possible to make further com-parisons to old analyses and results.

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The CFD analysis was done using a sector model describing cyclicallysymmetric sectors of stator 1 through stator 3. For the interfaces betweenthe stages to work, all stages need to have the same angle. Since each stageis made up by an integer number of blades, some scaling is required to fit theblades into the chosen sector angle. In this case the 45° angle was chosen inorder to preserve the geometry of rotor 2, which is the one being analyzedin this case. All other sectors are slightly scaled which means that the CFDmodel represents a turbine with a somewhat different number of blades. Aslong as this is taken into account when analyzing frequency contents, thescaling is not considered to affect results significantly in this case.

Two additional aerodynamic load cases were prepared based on a pre-vious study on RT forced response using VOLSOL . One case is based onthe raw VOLSOL output for the same blade here analyzed in ANSYS CFX. This data needs to be put through a fourier transform and mapped ontothe FE mesh before being read into ANSYS Mechanical for forced responseanalysis. The final case is based on the ANSYS Mechanical input preparedduring the previous study, allowing for an evaluation of the fourier transformand mapping methods.

The structural model used was built of around 90000 10-noded tetrahe-dron elements named SOLID187 in ANSYS Mechanical , see Figure 8. Itdescribes a one blade cyclically symmetric sector of the turbine. Since themodel is meant to be used for analyses of the second rotor only, the first rotorhas been approximated by a coarse block to increase computational efficiency.

Since AROMA can only handle a limited number of nodal DOFs, thepreviously described model had to be reduced before it could be processed.This step resulted in a 2912 element model describing only the second rotor.The reduced model can be viewed in Figure 9.

Material data input for the analysis was taken from a data file for theInconel alloy 718 used in previous analyses for RT. The damping ratio forthe entire structure was set to a value obtained from measurements and CFDduring a previous study.

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Figure 8: Cyclic sector model plotted in ANSYS Mechanical .

Figure 9: Reduced model plotted in ANSYS Mechanical .

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6.3 CFD Analysis

Input for aerodynamic loading is required for analysis in both ANSYS Me-chanical and AROMA . The goal here is to calculate how the pressure distri-bution on the rotor blades varies as the turbine spins. This was done througha CFD simulation where pressure distribution was registered with small timeincrements.

The analysis was done entirely in ANSYS CFX . Sliding interfaces wereused to simulate turbine rotation and periodic boundary conditions were seton the cyclic interfaces. Inlet and outlet pressure and temperature were setaccording to available RT data. The flow was then allowed to converge toa stable periodic pattern with regards to Rotor 1 and Rotor 2 blade forces.The timestep size was set to yield 100 timesteps per 45° sector passing andthe absolute pressure (pressure relative to absolute vacuum) was stored ateach step, resulting in a sampling frequency at 467300 Hz.

In post-processing, a script was created to export the nodal pressures onthe surface of the fourth blade (R2 blade 4) at every timestep. The resultingfiles provided a time history of the pressure at each surface node, which couldthen be further processed for frequency domain results.

6.4 Fourier Transform

To analyze the stresses that occur in the structure due to the excitationcaused at a specific frequency by Stator 3, the amplitude of this dynamicpressure frequency content at each node needs to be extracted. Through theCFD analysis, a pressure time history for each surface node was available.By applying a fourier transform, a pressure frequency spectrum could be cal-culated showing the amplitudes of the relevant frequency at each node.

For this purpose, a MATLAB routine was written making use of thebuilt in discrete fourier transform algorithm. The routine reads in outputfrom ANSYS CFX and sorts it into an array. The pressure time history foreach node is then extracted and an FFT is applied. The spectrum is plottedwith the frequency axis converted to EOs and relevant EOs marked to checkthe accuracy. Finally, the nodal pressures at the EO of interest is written toa file containing CFD mesh nodal frequency content. By running the AN-SYS CFX output previously obtained, a spectrum was obtained which canbe viewed in Figure 10. The spectrum shows the frequency response of everynode in order to show all excitations occuring in the structure. In this case

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the analysis focuses on the EO corresponding to 2S3, which was exported forthe next step of the process.

Figure 10: ANSYS CFX frequency spectrum with engine orders plotted inMATLAB

In order to be able to analyze the results produced in VOLSOL duringthe previous study, a similar code was written to process VOLSOL output.The same FFT method as in the above case was applied, and the resultingspectrum can be seen in Figure 11.

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Figure 11: VOLSOL frequency spectrum with engine orders plotted in MAT-LAB

6.5 Mapping of Loads

The data containing the aerodynamic loading cannot be directly applied tothe structural model. Since the meshes used for CFD and FE simulationsare not the same, the nodal pressures from the CFD analysis need to bemapped onto the corresponding nodes on the FE mesh. As the meshes candiffer significantly in structure and coarseness, an accurate mapping can bea difficult and time-consuming process.

In this case, a mapping routine developed internally at GKN Aerospacewas chosen for the CFD to FE pressure transfer. Rather than relying onshortest distance pressure mappings in three-dimensional space, the methodmaps on simplified Coon’s patched surfaces and identifies 2D positions onthe simplified surfaces for increased reliability and precision [5]. This givesgreater accuracy on the edges of the blade and gives the user more controlover the process. The final result is an array with real and imaginary forcecomponents on coordinate system axes which can be used for forced response.

Contour plots before and after mapping were inspected to make sure that

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the mapping procedure had not distorted the pressure distribution. As seenin Figure 12, details in the pressure distribution on the trailing edge werewell preserved onto the FE mesh. The mapped pressures are used to generatethe nodal forces needed for forced response analysis.

(a) CFX output (b) FE mesh mapping

Figure 12: Real pressure components on the trailing edge of the blade beforeand after mapping with identical contour plot settings

As the fine and coarse FE meshes share the exact same geometry andaxis of rotation, the load mapping from fine to coarse mesh was done usingan automated script written in MATLAB . A code was written that cyclesthrough the nodes on the fine mesh and adds the nodal forces to the nearestnode on the coarse mesh. The mapping path distances as well as total bladeforce are checked manually to make sure that no inconsistencies appear. Theresulting array is then written to a file that matches the input format ofeither ANSYS Mechanical or AROMA .

6.6 Forced Response in ANSYS Mechanical

With the nodal force amplitude and phases on the FE mesh available, thenext step in the procedure is a harmonic forced response analysis of the

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structure. ANSYS Mechanical 15.0 tools for cyclic symmetric analysis wereused with commands as specified in the following sections. Four differentcases were analyzed to compared different loads and FE models as seen inTable 1.

Case Model Loading1 Coarse ANSYS CFX2 Fine ANSYS CFX3 Fine VOLSOL4 Fine From previous study

Table 1: The CFD model sectors, plotted in ANSYS CFX

6.6.1 Prestress Conditions

Before starting the actual forced response analysis, the prestressed conditionof the structure needs to be calculated. This step takes the initial stressesincluding static pressure loads, thermal loads, and rotation into account inorder to define an initial state for the subsequent analyses. In this case,only a temperature load at 293 K was applied with tunif in order to getconsistency across all results.

6.6.2 Modal Analysis

The modal analysis was conducted using the perturbation method in ANSYSMechanical , a method that takes into account for example the stress stiff-ening effect. The frequency interval of the modal analysis was set to extract10 modes with the modopt command and nodal diameter corresponding to2S3 was selected with the cycopt command. The loads were read in this stepusing input files.

6.6.3 Harmonic analysis

To finally calculate the dynamic response of the structure, the mode su-perposition harmonic analysis was applied. This method uses the naturalfrequencies and modeshapes from the modal analysis and is initialized by se-lecting mode superposition with the hropt option. In this case, the frequencyspectrum was set between 20000 and 22000 Hz with 400 steps.

In the post-processing, the frequency response of a selected node wasthen written to an output file. Here a node on the mid point of the trailingedge was chosen as this is were the largest deformations occur for the mode

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being excited. The selected node on the fine and coarse mesh can be seen inFigure 13.

(a) Fine mesh (b) Coarse mesh

Figure 13: Renderings showing the nodes where forced response amplitudesare collected

With the harmonic analysis complete, the resulting nodal deflection out-put files were finally read and plotted using MATLAB .

6.7 Forced Response in AROMA

6.7.1 Pre-processing

Before running an analysis in AROMA , some pre-processing is required toproperly define the model. Since the program is still in a development phase,this was done in cooperation with Mauricio Guiterrez who is currently work-ing on the AROMA code. The first step was to prepare the assembly bycreating element components for the disk and blade parts of the structureusing ANSYS Mechanical . The components can be viewed in Figure 14. Forthis model it was not possible to get a clean cut on the interface due to thestructuring of the elements, and a small protrusion is visible at the root ofthe blade.

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Figure 14: Splitting the assembly into blade and disk components

After the components have been defined and saved, they can be usedto create node lists needed in AROMA . In order for the code to properlyinterpret the model, node lists are needed for the cyclic interfaces and in-terior nodes for the blade and disk components. A step by step instructionis meant to guide the user through the pre-processing, and was followed asclosely as possible. In this case, during pre-processing in ANSYS Mechanicalit was not possible to use the *status,cyclic xref n command as specified inthe instructions. This was probably due to the use of SOLID187 elementsin the model. This meant that the matching of cyclic interface nodes on theupper and lower side had to be done manually for the RT analysis.

The last step before running the analysis is to generate geometry, mass,and stiffness input for AROMA . This data was obtained through the samestatic analysis as the one applied during analysis in ANSYS Mechanical .This ensured that the forced response analysis in AROMA was based on thesame input as the results from ANSYS Mechanical .

6.7.2 Initiation and Settings

The AROMA user interface is based on a sequence of options and input pathsthat decide the type of analysis performed. Since it can become tedious giv-ing the same series of inputs every time one option needs to be changed,AROMA is usually run with a script containing the desired sequence of in-puts.

AROMA contains a built-in mapping algorithm that allows the user to

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input an array with nodal forces on the CFD mesh which is then trans-ferred automatically to the FE mesh used by the code. The built-in mappinginitially showed some problems when applied to the RT meshes. This waslikely related to a change in axis of rotation between CFD and FE modelswhich had not previously been encountered in earlier test cases. In order toget an accurate results comparison, the array of nodal forces prepared for thecoarse mesh analysis in ANSYS Mechanical was rewritten to match AROMAformatting, bypassing the built-in mapping in AROMA . This means thatresults from the two methods were based on the exact same forces.

To achieve good balance between accuracy and computation time, AROMAlets the user decide the number of modes used for the forced response anal-ysis. Generally a higher number of modes means higher accuracy, but sincethe most influential modeshapes are the lowest ones, increasing the numberof modes beyond a certain point leads to rapidly diminishing returns regard-ing results. The analysis was run with the numbers of modes ranging from1 to 200, showing that 100 modes was sufficient for good accuracy as seen inFigure 21 in the results section. A comparison was also done to assess differ-ence between Craig-Bampton and Craig-Chang reductions. As seen in Figure22, the comparison yielded no noticeable differences and Craig-Bampton wasused for the final results.

The same damping factor was used as in the ANSYS Mechanical analysis.The forced response was done on the same frequency interval and with thesame step size to give fair comparisons. Finally, a node number that corre-sponds to the same node as previously analyzed was used when extractingdeflections. The script was set to create an output file with deflections thatcould be used when creating comparison plots in MATLAB .

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7 Results

7.1 Overview

Figure 15 shows an overview of results from all performed analyses withmodel density and aerodynamic loading as specified in the plot. AROMAresults are based on Craig-Bampton reduction and 100 modes.

Figure 15: Overview of results

7.2 Specific Comparisons

7.2.1 ANSYS CFX and VOLSOL

Figure 16 presents a comparison plot showing differences between resultsbased on VOLSOL and ANSYS CFX output on the fine model. Below,Figure 17 and Figure 18 shows absolute pressure amplitudes of 2S3 based onmapped ANSYS CFX and VOLSOL output, with flow direction from leftto right in plots. Note that VOLSOL pressure scale goes 67 % higher thanANSYS CFX .

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Figure 16: VOLSOL and ANSYS CFX comparison

(a) Suction surface (b) Pressure surface

Figure 17: Absolute pressure amplitudes, ANSYS CFX

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(a) Suction surface (b) Pressure surface

Figure 18: Absolute pressure amplitudes, VOLSOL

7.2.2 Fine and Coarse Mesh

Figure 19 shows differences between results based on ANSYS CFX outputapplied to fine and coarse models.

Figure 19: Fine and coarse mesh comparison

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7.2.3 FFT and Mapping

Figure 20 presents a comparison plot showing differences between resultsbased on VOLSOL output processed for this report and output processed inthe previous study.

Figure 20: FFT and mapping comparison

7.2.4 AROMA Settings

Figure 21 shows the effect of increasing number of used modes with Craig-Bampton reduction. A zoomed in view is provided for details. Below, Figure22 shows a comparison based on Craig-Bampton and Craig-Chang reduc-tions.

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(a) Regular view (b) Zoomed in view

Figure 21: Increasing number of used modes

(a) Regular view (b) Zoomed in view

Figure 22: Reduction methods

7.2.5 ANSYS Mechanical and AROMA

Figure 23 shows a comparison between results produced in AROMA and AN-SYS Mechanical . The AROMA result is based on Craig-Bampton reductionand 100 modes.

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Figure 23: Forced response tool comparison

7.3 Test Data Comparison

When put through HCF analysis, the deflections acquired during the oldstudy produced stresses that were smaller than test data stresses by a factorranging between 3.5 and 5.1. The stresses based on ANSYS CFX output andnew FFT and mapping methods showed an increase by a factor 3.3 comparedto the old results [8].

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8 Conclusions

8.1 On Result Interpretation

When reviewing results presented in this report, it is important to keep inmind that all numbers outside of the test data comparison only representsdeflections. While the plots and ratios give a good approximation, actualdifferences in material stresses require a complete HCF analysis. This typeof result is only presented for the fine mesh ANSYS CFX simulation in theform of a comparison ratio to results of the previous study and test results.

8.2 CFD Methods

As seen in Figure 16, VOLSOL shows an increase in deflection by 11 % overANSYS CFX results. However, looking at the pressure amplitudes over theblade surfaces in Figure 17 and 18, the differences are significant. VOLSOLyields a maximum amplitude that is 68 % higher than the maximum fromANSYS CFX . The pressure distribution also shows large differences. TheVOLSOL output shows a high pressure region at the trailing edge, whichis reasonable since the excitations at this frequency should mainly originatefrom Stator 3 which sits directly behind Rotor 2. ANSYS CFX output showsa more uneven distribution, with a high pressure area at the leading edge aswell. There is also more scattering in the pressure amplitudes on the surfaces.

Despite these differences, the forced response results are relatively sim-ilar. This could be due to the fact that both CFD outputs have similarpressure amplitudes at the midpoint of the trailing edge where the mode hasits maximum deflection. Investigating other modeshapes could potentiallyyield large differences if the maximum deflection is at a point where ANSYSCFX and VOLSOL shows stronger variations, such as the leading edge.

8.3 Mesh Density Reduction

As seen in Figure 19, reducing the mesh density of the FE model notablyaffects results. The coarse model is less flexible showing a 12 % smaller de-flection, and the resonant frequency has increased by 690 Hz or 3%. Makingan FE model less detailed should be avoided when possible.

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8.4 FFT and CFD to FE Mesh Mapping

Figure 20 shows a 229 % increase in deflection from the previous study to thecurrent VOLSOL analysis. The FFT and mapping process are a part of thisdifference, which highlights the need for precise and reliable methods for thisstep of the forced response analysis. The exact sources of the inconsistencieshave not been investigated further in this report, but would make a goodsubject for future studies.

Mapping loads between different meshes remains a problem. There iscurrently no established standard method at GKN Aerospace for the map-ping procedure, which could be a potential source of errors when differentmethods are used in different projects. In previous RT forced response anal-yses, the mapping was based on an averaging process where pressures fromCFD mesh nodes where weighted based on distance to the FE mesh nodeand then applied. This could lead to inaccuracies in regions on the surfacewith a steep pressure gradient such as the leading edge where high pressurerisks being spread across a larger area.

To ensure that no inconsistencies in results arise from the force mappingstep, a design practice outlining a verified mapping method would be the bestsolution. The method developed at GKN Aerospace is a good candidate, asit has already been tested and verified for several cases (G. Hogstrom, per-sonal communication, December 8, 2014). By developing the process throughfurther test cases and possibly making it more intuitive for new users, thetool could be established as a standard method to reduce errors in analysiscases where this type of mapping is required.

8.5 Test Data Comparison

A comparison of available test data and analysis results from the previousstudy showed that analysis stresses were lower than test data by a factorranging from 3.5 to 5.1. This is significant, since there is a risk that theresonance is ignored and ”flies under the radar” if the simulation cannotshow that it is close to causing a blade failure. The results in this thesisshould ideally have showed an increase by a factor in the 3.5-5.1 intervalto provide total accuracy, instead they fell slightly short at a factor 3.1.However, these results would still be significant enough to warrant furtheranalysis during the design process. The test data comparison gives an overallvalidation that the methods applied in this work are a notable improvement

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on methods applied during the previous study.

8.6 AROMA Evaluation

8.6.1 Analysis Results

The reduction methods showed no notable differences for this case, as seenin Figure 22. 100 modes was sufficient regarding accuracy as seen in Figure21. With a frequency increase of 60 Hz or 0.3 % and a deflection increaseof 18 %, the results stayed relatively close to the ANSYS Mechanical resultsas seen in Figure 23. Craig-Bampton and Craig-Chang show nearly identicalresults for this case.

8.6.2 Usability

Working with the AROMA tool is still fairly unintuitive compared to estab-lished forced response tools such as ANSYS Mechanical . The pre-processingcan be difficult to carry out properly for a new user, as the important com-ponent splitting is currently done manually using ANSYS Mechanical . Forthe RT model, the pre-processing steps outlined in the instruction manualchecklist could not be followed exactly due to the type of element used inthe model. This was a minor issue, but highlights the fact that working withAROMA still requires a very good understanding of how the code works. Ifthe code is to be used within industry applications, it needs to be made morerobust and consistent across different types of FE-models.

It is important to point out that the AROMA analyses carried out forthis report did not save any time compared to the traditional method withANSYS Mechanical . The original dense model took over an hour to analyzein ANSYS Mechanical , but was too dense for the AROMA code to handle.After the model had been reduced to the coarse version used in AROMA ,it could be analyzed in ANSYS Mechanical in a matter of seconds. In orderfor the reduced order modeling of AROMA to give any noticeable speedimprovements, the code needs to be able to handle much denser modelswhere the DOF reduction can make a meaningful difference.

8.6.3 Verdict

AROMA shows promise, but in its current form it is far from introductionin industry applications. While the ability of the code to produce accurateforced response results with ROM has been proven for the RT test case,several issues still need to be addressed. To reduce the risk of errors, the

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component splitting process needs to be made more versatile and compatiblewith different types of FE model element types. In order to make full useof the reduced order modeling, AROMA needs to be able to handle muchdenser FE models. Not only does reducing the level of detail in the FE modelreduce accuracy, but it essentially eliminates the codes main advantage whichis speed.

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9 Acknowledgments

The paper you are reading would not have been possible without the fol-lowing: Thank you to Clas Andersson for support and help with structuralanalysis, Pieter Groth for help and valuable input on CFD analysis, StaffanBrodin for support and encouragement, Gunnar Hogstrom for help with map-ping of loads, Mauricio Gutierrez for support and guidance on AROMA , andNiklas Edin for help on sorting out results from previous RT studies.

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References

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[2] Guyan, R. J., Reduction of Stiffness and Mass Matrices. AIAA Journal,Vol. 3, No. 2, pp. 380, 1965.

[3] Craig, R.R,. Chang,C.-J., A Review of Substructure Coupling Methodsfor Dynamic Analysis. NASA CP-2001, Washington, DC, pp 393-408,1976.

[4] Gutierrez, M., Gezork, T., Yang, S., Fransson T. H., Andersson, C.,Vogt, D. M., Forced Response Analysis Of A Transonic Turbine Us-ing A Free Interface Component Mode Synthesis Method. Heat andPower Technology/THRUST, KTH, Stockholm, Sweden, Departmentof Fatigue, GKN Aerospace, Trollhattan, Sweden, ITSM - Instituteof Thermal Turbomachinery and Machinery Laboratory, University ofStuttgart, Stuttgart, Germany

[5] Hogstrom, G., An Ansys macro library for mapping of Gas PressureData from a file to an Ansys FE model node surface. VOLS:10173659

[6] Wildheim S. J., Excitation of Rotationally Periodic Structures, Journalof Applied Mechanics, Vol 46 No 4 December 1979, p 878-882

[7] Johansson A., Internal report on reference turbine, VOLS: 10064510-01

[8] Andersson C., Harmonic Forced Response in ANSYS 15.0 EmployingCyclic Symmetry, VOLS:10208743-01

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